# Question 33661: Frobenius Fixes Every Element of This Prime Field

[Return to catalogue page 169](https://ocme.evemisslab.com/en/questions/page/169/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261031-r06-finite-field-foundations/#q-20261014-advance-20261031-r06-finite-field-0086)

This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0066-33661 · q-20261014-advance-20261031-r06-finite-field-0086



## Original question and givens

Use the complete prime field Z_31, with prime p=31, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=10, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.


- G001 — Use the complete prime field Z_31, with prime p=31, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=10, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-finite-field](https://ocme.evemisslab.com/en/mko/mko-finite-field/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-mod31-fielddeep-frobenius, [domain, element, operation, p]]

After

derived = [fielddeep-frobenius, Exact_prime_field_Zp]

Conditions: none

THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged.

Step identifiers and dependenciesRule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001





S002 · impliesBefore

source roles = [31, [2, 3, 4, 5]]

After

derived = [[1, 1, 3, 1], Yes]

Conditions: none

Trial EVERY divisor2 through floor(sqrt(p));a composite would have a factor there.The empty test for p=2 is valid.This qualifies THIS prime field Z_p,not every finite field’s order or every finite ring as a field.

Step identifiers and dependenciesRule: Field_PRIME_complete_trial_divisor_qualification · Uses: G001, S001





S003 · calculationBefore

Field ORIGINAL previous power times element k=1 := mul(1, 10)

After

Field ORIGINAL previous power times element k=1 = 10

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

source roles = [power exponent1, 10, 31]

After

derived = [0, 10]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S003





S005 · calculationBefore

Field ORIGINAL previous power times element k=2 := mul(10, 10)

After

Field ORIGINAL previous power times element k=2 = 100

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

source roles = [power exponent2, 100, 31]

After

derived = [3, 7]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S005





S007 · calculationBefore

Field ORIGINAL previous power times element k=3 := mul(7, 10)

After

Field ORIGINAL previous power times element k=3 = 70

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · impliesBefore

source roles = [power exponent3, 70, 31]

After

derived = [2, 8]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S007





S009 · calculationBefore

Field ORIGINAL previous power times element k=4 := mul(8, 10)

After

Field ORIGINAL previous power times element k=4 = 80

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · impliesBefore

source roles = [power exponent4, 80, 31]

After

derived = [2, 18]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S009





S011 · calculationBefore

Field ORIGINAL previous power times element k=5 := mul(18, 10)

After

Field ORIGINAL previous power times element k=5 = 180

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · impliesBefore

source roles = [power exponent5, 180, 31]

After

derived = [5, 25]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S011





S013 · calculationBefore

Field ORIGINAL previous power times element k=6 := mul(25, 10)

After

Field ORIGINAL previous power times element k=6 = 250

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · impliesBefore

source roles = [power exponent6, 250, 31]

After

derived = [8, 2]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S013





S015 · calculationBefore

Field ORIGINAL previous power times element k=7 := mul(2, 10)

After

Field ORIGINAL previous power times element k=7 = 20

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · impliesBefore

source roles = [power exponent7, 20, 31]

After

derived = [0, 20]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S015





S017 · calculationBefore

Field ORIGINAL previous power times element k=8 := mul(20, 10)

After

Field ORIGINAL previous power times element k=8 = 200

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · impliesBefore

source roles = [power exponent8, 200, 31]

After

derived = [6, 14]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S017





S019 · calculationBefore

Field ORIGINAL previous power times element k=9 := mul(14, 10)

After

Field ORIGINAL previous power times element k=9 = 140

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018





S020 · impliesBefore

source roles = [power exponent9, 140, 31]

After

derived = [4, 16]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S019





S021 · calculationBefore

Field ORIGINAL previous power times element k=10 := mul(16, 10)

After

Field ORIGINAL previous power times element k=10 = 160

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020





S022 · impliesBefore

source roles = [power exponent10, 160, 31]

After

derived = [5, 5]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S021





S023 · calculationBefore

Field ORIGINAL previous power times element k=11 := mul(5, 10)

After

Field ORIGINAL previous power times element k=11 = 50

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022





S024 · impliesBefore

source roles = [power exponent11, 50, 31]

After

derived = [1, 19]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S023





S025 · calculationBefore

Field ORIGINAL previous power times element k=12 := mul(19, 10)

After

Field ORIGINAL previous power times element k=12 = 190

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024





S026 · impliesBefore

source roles = [power exponent12, 190, 31]

After

derived = [6, 4]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S025





S027 · calculationBefore

Field ORIGINAL previous power times element k=13 := mul(4, 10)

After

Field ORIGINAL previous power times element k=13 = 40

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026





S028 · impliesBefore

source roles = [power exponent13, 40, 31]

After

derived = [1, 9]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S027





S029 · calculationBefore

Field ORIGINAL previous power times element k=14 := mul(9, 10)

After

Field ORIGINAL previous power times element k=14 = 90

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028





S030 · impliesBefore

source roles = [power exponent14, 90, 31]

After

derived = [2, 28]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S029





S031 · calculationBefore

Field ORIGINAL previous power times element k=15 := mul(28, 10)

After

Field ORIGINAL previous power times element k=15 = 280

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030





S032 · impliesBefore

source roles = [power exponent15, 280, 31]

After

derived = [9, 1]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S031





S033 · calculationBefore

Field ORIGINAL previous power times element k=16 := mul(1, 10)

After

Field ORIGINAL previous power times element k=16 = 10

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032





S034 · impliesBefore

source roles = [power exponent16, 10, 31]

After

derived = [0, 10]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S033





S035 · calculationBefore

Field ORIGINAL previous power times element k=17 := mul(10, 10)

After

Field ORIGINAL previous power times element k=17 = 100

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034





S036 · impliesBefore

source roles = [power exponent17, 100, 31]

After

derived = [3, 7]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S035





S037 · calculationBefore

Field ORIGINAL previous power times element k=18 := mul(7, 10)

After

Field ORIGINAL previous power times element k=18 = 70

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036





S038 · impliesBefore

source roles = [power exponent18, 70, 31]

After

derived = [2, 8]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S037





S039 · calculationBefore

Field ORIGINAL previous power times element k=19 := mul(8, 10)

After

Field ORIGINAL previous power times element k=19 = 80

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038





S040 · impliesBefore

source roles = [power exponent19, 80, 31]

After

derived = [2, 18]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S039





S041 · calculationBefore

Field ORIGINAL previous power times element k=20 := mul(18, 10)

After

Field ORIGINAL previous power times element k=20 = 180

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040





S042 · impliesBefore

source roles = [power exponent20, 180, 31]

After

derived = [5, 25]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S041





S043 · calculationBefore

Field ORIGINAL previous power times element k=21 := mul(25, 10)

After

Field ORIGINAL previous power times element k=21 = 250

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042





S044 · impliesBefore

source roles = [power exponent21, 250, 31]

After

derived = [8, 2]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S043





S045 · calculationBefore

Field ORIGINAL previous power times element k=22 := mul(2, 10)

After

Field ORIGINAL previous power times element k=22 = 20

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044





S046 · impliesBefore

source roles = [power exponent22, 20, 31]

After

derived = [0, 20]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S045





S047 · calculationBefore

Field ORIGINAL previous power times element k=23 := mul(20, 10)

After

Field ORIGINAL previous power times element k=23 = 200

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046





S048 · impliesBefore

source roles = [power exponent23, 200, 31]

After

derived = [6, 14]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S047





S049 · calculationBefore

Field ORIGINAL previous power times element k=24 := mul(14, 10)

After

Field ORIGINAL previous power times element k=24 = 140

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048





S050 · impliesBefore

source roles = [power exponent24, 140, 31]

After

derived = [4, 16]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S049





S051 · calculationBefore

Field ORIGINAL previous power times element k=25 := mul(16, 10)

After

Field ORIGINAL previous power times element k=25 = 160

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050





S052 · impliesBefore

source roles = [power exponent25, 160, 31]

After

derived = [5, 5]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S051





S053 · calculationBefore

Field ORIGINAL previous power times element k=26 := mul(5, 10)

After

Field ORIGINAL previous power times element k=26 = 50

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052





S054 · impliesBefore

source roles = [power exponent26, 50, 31]

After

derived = [1, 19]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S053





S055 · calculationBefore

Field ORIGINAL previous power times element k=27 := mul(19, 10)

After

Field ORIGINAL previous power times element k=27 = 190

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054





S056 · impliesBefore

source roles = [power exponent27, 190, 31]

After

derived = [6, 4]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S055





S057 · calculationBefore

Field ORIGINAL previous power times element k=28 := mul(4, 10)

After

Field ORIGINAL previous power times element k=28 = 40

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056





S058 · impliesBefore

source roles = [power exponent28, 40, 31]

After

derived = [1, 9]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S057





S059 · calculationBefore

Field ORIGINAL previous power times element k=29 := mul(9, 10)

After

Field ORIGINAL previous power times element k=29 = 90

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058





S060 · impliesBefore

source roles = [power exponent29, 90, 31]

After

derived = [2, 28]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S059





S061 · calculationBefore

Field ORIGINAL previous power times element k=30 := mul(28, 10)

After

Field ORIGINAL previous power times element k=30 = 280

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S060





S062 · impliesBefore

source roles = [power exponent30, 280, 31]

After

derived = [9, 1]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S061





S063 · calculationBefore

Field ORIGINAL previous power times element k=31 := mul(1, 10)

After

Field ORIGINAL previous power times element k=31 = 10

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S062





S064 · impliesBefore

source roles = [power exponent31, 10, 31]

After

derived = [0, 10]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S063





S065 · impliesBefore

source roles = [31, 10, [10, 7, 8, 18, 25, 2, 20, 14, 16, 5, 19, 4, 9, 28, 1, 10, 7, 8, 18, 25, 2, 20, 14, 16, 5, 19, 4, 9, 28, 1, 10]]

After

derived = [10, Yes]

Conditions: none

Repeated multiplication checks element^p=element in THIS prime field,including zero.Frobenius need not fix every element of an arbitrary extension field;do not widen the conclusion.

Step identifiers and dependenciesRule: Field_FROBENIUS_prime_field_INCLUDES_zero · Uses: G001, S064









## Reverse checks


- S065 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S064 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S063 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=31 with exact rational arithmetic, not rounded decimals.
- S062 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S061 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=30 with exact rational arithmetic, not rounded decimals.
- S060 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S059 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=29 with exact rational arithmetic, not rounded decimals.
- S058 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S057 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=28 with exact rational arithmetic, not rounded decimals.
- S056 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S055 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=27 with exact rational arithmetic, not rounded decimals.
- S054 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S053 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=26 with exact rational arithmetic, not rounded decimals.
- S052 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S051 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=25 with exact rational arithmetic, not rounded decimals.
- S050 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S049 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=24 with exact rational arithmetic, not rounded decimals.
- S048 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S047 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=23 with exact rational arithmetic, not rounded decimals.
- S046 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S045 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=22 with exact rational arithmetic, not rounded decimals.
- S044 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S043 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=21 with exact rational arithmetic, not rounded decimals.
- S042 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S041 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=20 with exact rational arithmetic, not rounded decimals.
- S040 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S039 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=19 with exact rational arithmetic, not rounded decimals.
- S038 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S037 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=18 with exact rational arithmetic, not rounded decimals.
- S036 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S035 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=17 with exact rational arithmetic, not rounded decimals.
- S034 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S033 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=16 with exact rational arithmetic, not rounded decimals.
- S032 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S031 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=15 with exact rational arithmetic, not rounded decimals.
- S030 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S029 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=14 with exact rational arithmetic, not rounded decimals.
- S028 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S027 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=13 with exact rational arithmetic, not rounded decimals.
- S026 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S025 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=12 with exact rational arithmetic, not rounded decimals.
- S024 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S023 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=11 with exact rational arithmetic, not rounded decimals.
- S022 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S021 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=10 with exact rational arithmetic, not rounded decimals.
- S020 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S019 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=9 with exact rational arithmetic, not rounded decimals.
- S018 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S017 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=8 with exact rational arithmetic, not rounded decimals.
- S016 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S015 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=7 with exact rational arithmetic, not rounded decimals.
- S014 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S013 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=6 with exact rational arithmetic, not rounded decimals.
- S012 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S011 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=5 with exact rational arithmetic, not rounded decimals.
- S010 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S009 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=4 with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S007 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=3 with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S005 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=2 with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S003 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=1 with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- S001 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.





## Result

(10, 31, 10, Yes, Frobenius_on_prime_field)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 3db9f832bf3869350c2276e91ac2ab884e44a1a0cc0ebbc0bcdb8bfcfc318f3c
- candidate record SHA-256: 5af1a212a82b9c163cc9c0f44b5b20b10d32346ff325c7ef2de87252324c26b3
- semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267
- question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261031-r06-finite-field-0086.json)