Question 15125: Field Division Uses a Nonzero Denominator and the Same Table

Original question and givens

Use the same already qualified complete finite commutative unital-ring tables on Fin(5)={0,…,4}, addition=[[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]] and multiplication=[[0, 1, 2, 3, 4], [1, 1, 1, 1, 1], [2, 1, 4, 0, 3], [3, 1, 0, 4, 2], [4, 1, 3, 2, 0]] in carrier-label order. The actual zero and one are distinct and need not be labels 0 and 1. The inverse audit is additional to, not a replacement for, those ring axioms. Division by zero is never defined. A finite field has prime-power order, not necessarily prime order. Under field qualification, numerator=1, denominator=0 is nonzero. Give (numerator,denominator,inverse,quotient,check,pass?).

• G001 — Use the same already qualified complete finite commutative unital-ring tables on Fin(5)={0,…,4}, addition=[[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]] and multiplication=[[0, 1, 2, 3, 4], [1, 1, 1, 1, 1], [2, 1, 4, 0, 3], [3, 1, 0, 4, 2], [4, 1, 3, 2, 0]] in carrier-label order. The actual zero and one are distinct and need not be labels 0 and 1. The inverse audit is additional to, not a replacement for, those ring axioms. Division by zero is never defined. A finite field has prime-power order, not necessarily prime order. Under field qualification, numerator=1, denominator=0 is nonzero. Give (numerator,denominator,inverse,quotient,check,pass?).

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

same complete carrier Fin(5);addition=[[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]];multiplication=[[0, 1, 2, 3, 4], [1, 1, 1, 1, 1], [2, 1, 4, 0, 3], [3, 1, 0, 4, 2], [4, 1, 3, 2, 0]]

actual zero=1;one=0;additive inverses=[4, 1, 3, 2, 0];additive commute/associate=[Yes, Yes];multiplicative commute/associate=[Yes, Yes];left/right distributive=[Yes, Yes]

element label=0;actual zero=1;one=0;two-sided multiplicative inverse candidates=[0]

selected inverse witness=0;nonzero inverse obligation=Yes

element label=1;actual zero=1;one=0;two-sided multiplicative inverse candidates=[]

selected inverse witness=null;nonzero inverse obligation=No

element label=2;actual zero=1;one=0;two-sided multiplicative inverse candidates=[3]

selected inverse witness=3;nonzero inverse obligation=Yes

element label=3;actual zero=1;one=0;two-sided multiplicative inverse candidates=[2]

selected inverse witness=2;nonzero inverse obligation=Yes

element label=4;actual zero=1;one=0;two-sided multiplicative inverse candidates=[4]

selected inverse witness=4;nonzero inverse obligation=Yes

qualified field;raw numerator label=1;nonzero denominator label=0;inverse=0;multiplication row=[1, 1, 1, 1, 1]

quotient label=1;quotient*denominator=1=numerator

Reverse checks

• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(1, 0, 0, 1, 1, Yes)

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.