Question 33657: Frobenius Fixes Every Element of This Prime Field

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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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

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001

S002 · implies

Before

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 dependencies

Rule: Field_PRIME_complete_trial_divisor_qualification · Uses: G001, S001

S003 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S003

S005 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S005

S007 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S007

S009 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S009

S011 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S011

S013 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S013

S015 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S015

S017 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S017

S019 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S019

S021 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S021

S023 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S023

S025 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · implies

Before

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S025

S027 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S027

S029 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · implies

Before

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S029

S031 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · implies

Before

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S031

S033 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · implies

Before

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S033

S035 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · implies

Before

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S035

S037 · implies

Before

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 dependencies

Rule: 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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