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Question 33631: Characteristic Is the Additive Order of One

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0066-33631 · q-20261014-advance-20261031-r06-finite-field-0006

Original question and givens

Use the complete prime field Z_13, with prime p=13, 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. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.

  • G001 — Use the complete prime field Z_13, with prime p=13, 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. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.

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.

Step-by-step derivation

S001 · implies

Before

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

After

derived = [fielddeep-characteristic, 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 = [13, [2, 3]]

After

derived = [[1, 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 · implies

Before

source roles = [successive sum of one k=0, 0, 13]

After

derived = [0, 0]

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, S002

S004 · implies

Before

source roles = [successive sum of one k=1, 1, 13]

After

derived = [0, 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, S003

S005 · implies

Before

source roles = [successive sum of one k=2, 2, 13]

After

derived = [0, 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, S004

S006 · implies

Before

source roles = [successive sum of one k=3, 3, 13]

After

derived = [0, 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, S005

S007 · implies

Before

source roles = [successive sum of one k=4, 4, 13]

After

derived = [0, 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, S006

S008 · implies

Before

source roles = [successive sum of one k=5, 5, 13]

After

derived = [0, 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, S007

S009 · implies

Before

source roles = [successive sum of one k=6, 6, 13]

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, S008

S010 · implies

Before

source roles = [successive sum of one k=7, 7, 13]

After

derived = [0, 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, S009

S011 · implies

Before

source roles = [successive sum of one k=8, 8, 13]

After

derived = [0, 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, S010

S012 · implies

Before

source roles = [successive sum of one k=9, 9, 13]

After

derived = [0, 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, S011

S013 · implies

Before

source roles = [successive sum of one k=10, 10, 13]

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 dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S012

S014 · implies

Before

source roles = [successive sum of one k=11, 11, 13]

After

derived = [0, 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, S013

S015 · implies

Before

source roles = [successive sum of one k=12, 12, 13]

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, S014

S016 · implies

Before

source roles = [successive sum of one k=13, 13, 13]

After

derived = [1, 0]

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 · implies

Before

source roles = [13, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0]]

After

derived = [13]

Conditions: none

List sums of0 through p copies of1.Residues1..p−1 are nonzero and p returns0,so the FIRST POSITIVE additive return is p.Zero is not a positive return;this is not multiplicative order.

Step identifiers and dependencies

Rule: Field_CHARACTERISTIC_first_POSITIVE_additive_return · Uses: G001, S016

Reverse checks

  • S017 · 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.
  • 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 — 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.
  • 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 — 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.
  • 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 — 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.
  • 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 — 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.
  • 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 — 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.
  • 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 — 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.
  • 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 — 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.
  • 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

(13, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], 13, additive_order_of_one)

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: d2cac1cb67fb1773a126a28955cb01894d6df60852a80e7712b72f92a90f554d
  • candidate record SHA-256: 5d6884126fea3583da928119533d745531c2332712874f8ed8dc793c1ac16ef7
  • semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267
  • question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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