# Question 33657: Frobenius Fixes Every Element of This Prime Field

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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.

R0066-33657 · q-20261014-advance-20261031-r06-finite-field-0082



## Original question and givens

Use the complete prime field Z_17, with prime p=17, 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=6, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.


- G001 — Use the complete prime field Z_17, with prime p=17, 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=6, 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 = [17, [2, 3, 4]]

After

derived = [[1, 2, 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, 6)

After

Field ORIGINAL previous power times element k=1 = 6

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, 6, 17]

After

derived = [0, 6]

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(6, 6)

After

Field ORIGINAL previous power times element k=2 = 36

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, 36, 17]

After

derived = [2, 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, S005





S007 · calculationBefore

Field ORIGINAL previous power times element k=3 := mul(2, 6)

After

Field ORIGINAL previous power times element k=3 = 12

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, 12, 17]

After

derived = [0, 12]

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(12, 6)

After

Field ORIGINAL previous power times element k=4 = 72

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, 72, 17]

After

derived = [4, 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, S009





S011 · calculationBefore

Field ORIGINAL previous power times element k=5 := mul(4, 6)

After

Field ORIGINAL previous power times element k=5 = 24

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, 24, 17]

After

derived = [1, 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, S011





S013 · calculationBefore

Field ORIGINAL previous power times element k=6 := mul(7, 6)

After

Field ORIGINAL previous power times element k=6 = 42

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, 42, 17]

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, S013





S015 · calculationBefore

Field ORIGINAL previous power times element k=7 := mul(8, 6)

After

Field ORIGINAL previous power times element k=7 = 48

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, 48, 17]

After

derived = [2, 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, S015





S017 · calculationBefore

Field ORIGINAL previous power times element k=8 := mul(14, 6)

After

Field ORIGINAL previous power times element k=8 = 84

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, 84, 17]

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, S017





S019 · calculationBefore

Field ORIGINAL previous power times element k=9 := mul(16, 6)

After

Field ORIGINAL previous power times element k=9 = 96

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, 96, 17]

After

derived = [5, 11]

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(11, 6)

After

Field ORIGINAL previous power times element k=10 = 66

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, 66, 17]

After

derived = [3, 15]

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(15, 6)

After

Field ORIGINAL previous power times element k=11 = 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, S022





S024 · impliesBefore

source roles = [power exponent11, 90, 17]

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, S023





S025 · calculationBefore

Field ORIGINAL previous power times element k=12 := mul(5, 6)

After

Field ORIGINAL previous power times element k=12 = 30

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, 30, 17]

After

derived = [1, 13]

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(13, 6)

After

Field ORIGINAL previous power times element k=13 = 78

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, 78, 17]

After

derived = [4, 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, S027





S029 · calculationBefore

Field ORIGINAL previous power times element k=14 := mul(10, 6)

After

Field ORIGINAL previous power times element k=14 = 60

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, 60, 17]

After

derived = [3, 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, S029





S031 · calculationBefore

Field ORIGINAL previous power times element k=15 := mul(9, 6)

After

Field ORIGINAL previous power times element k=15 = 54

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, 54, 17]

After

derived = [3, 3]

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(3, 6)

After

Field ORIGINAL previous power times element k=16 = 18

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, 18, 17]

After

derived = [1, 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, S033





S035 · calculationBefore

Field ORIGINAL previous power times element k=17 := mul(1, 6)

After

Field ORIGINAL previous power times element k=17 = 6

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, 6, 17]

After

derived = [0, 6]

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 · impliesBefore

source roles = [17, 6, [6, 2, 12, 4, 7, 8, 14, 16, 11, 15, 5, 13, 10, 9, 3, 1, 6]]

After

derived = [6, 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, S036









## Reverse checks


- S037 · 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.
- 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

(6, 17, 6, 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: a96eb60b5b00b84a6a37b257bf75628d71d47d7570ba9bcc836a6b60f7f2b93e
- candidate record SHA-256: 3a6d05355146df5c703124b3422b794c26916d160e9b5dfe599bdbe3a096b851
- semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267
- question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc
- Lean: NOT_RUN · human review: NOT_PERFORMED

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