derivation-mko-finite-field-01 · ocme-new142-reader-adoption-20261021
F3 is a field and 2/2=1; nonzero multiplication maps supply inverses in a finite domain.
Complete F3 tables and a finite-domain lemma; explicit F4 only as the nonprime-order boundary.
calcG1 — F3={0,1,2}, with integer addition/multiplication reduced modulo 3. Also let D be a finite commutative unital integral domain with 1 distinct from 0.F_3 — field modulo 3D — arbitrary finite integral domainm_a — multiplication-by-a self-mapcalc · prerequisite — Assume ring arithmetic of integers and F2 polynomials, congruence arithmetic, and injective self-maps of a finite set being surjective; an integral domain has no zero divisors.mko_definition · cited — Use the complete finite ring tables to retain 1 distinct from 0, multiplicative commutativity and a two-sided inverse for every nonzero element. Division is defined only for nonzero denominators. A finite integral domain becomes a field through the finite injective-to-surjective multiplication-map theorem. A finite field has prime-power order, not necessarily prime order. · mko-finite-fieldBefore
F_3=\{0,1,2\}After
\begin{array}{c|ccc}+&0&1&2\\\hline 0&0&1&2\\1&1&2&0\\2&2&0&1\\\end{array}Conditions: none
Rule: calc · Uses: G1
Leaf basis: calc
All nine sums are listed and lie in the carrier. Zero is the additive identity, and each ordered pair has exactly one sum.
Before
\begin{array}{c|ccc}+&0&1&2\\\hline 0&0&1&2\\1&1&2&0\\2&2&0&1\\\end{array}After
-0=0,\quad -1=2,\quad -2=1
Conditions: none
Rule: calc · Uses: G1, S01
Leaf basis: calc
The table supplies every additive inverse; associativity and commutativity descend from integer addition.
Before
a,b\in F_3
After
\begin{array}{c|ccc}*&0&1&2\\\hline 0&0&0&0\\1&0&1&2\\2&0&2&1\\\end{array}Conditions: none
Rule: calc · Uses: G1, S02
Leaf basis: calc
All nine products are listed. Integer associativity and distributivity survive reduction, and the table is symmetric.
Before
\begin{array}{c|ccc}*&0&1&2\\\hline 0&0&0&0\\1&0&1&2\\2&0&2&1\\\end{array}After
1\ne0,\quad 1^{-1}=1,\quad2^{-1}=2Conditions: none
Rule: calc · Uses: G1, S03
Leaf basis: calc
The row of 1 fixes every element; both nonzero elements, 1 and 2, have two-sided inverses in the table.
Before
2/2:=2\cdot2^{-1}After
2/2=2\cdot2=1\quad(\bmod 3)
Conditions: denominator 2 is nonzero in F3
Rule: calc · Uses: G1, S04
Leaf basis: calc
The denominator 2 is nonzero, so division is multiplication by its inverse and gives 1.
Before
a\in D\setminus\{0\},\quad m_a(x)=axAfter
m_a(x)=m_a(y)\Rightarrow a(x-y)=0\Rightarrow x=y
Conditions: a is nonzero; D has no zero divisors
Rule: calc · Uses: G1
Leaf basis: calc
In the stated finite domain, absence of zero divisors and a nonzero a make its multiplication self-map injective.
Before
m_a:D\to D\text{ injective},\quad |D|<\inftyAfter
m_a\text{ surjective}\Rightarrow\exists b\in D:\ ab=1Conditions: none
Rule: calc · Uses: G1, S06
Leaf basis: calc
An injective finite self-map is surjective, so 1 has a preimage b; commutativity also gives ba=1.
Before
t^2=t+1\text{ over }F_2After
\begin{array}{c|cccc}+&0&1&t&t+1\\\hline 0&0&1&t&t+1\\1&1&0&t+1&t\\t&t&t+1&0&1\\t+1&t+1&t&1&0\\\end{array}\qquad\begin{array}{c|cccc}*&0&1&t&t+1\\\hline 0&0&0&0&0\\1&0&1&t&t+1\\t&0&t&t+1&1\\t+1&0&t+1&1&t\\\end{array}Conditions: none
Rule: calc · Uses: G1
Leaf basis: calc
The four remainders 0,1,t,t+1 reduce by t^2=t+1, giving both complete tables. Sums and products preserve differences divisible by t^2+t+1.
Before
\begin{array}{c|cccc}*&0&1&t&t+1\\\hline 0&0&0&0&0\\1&0&1&t&t+1\\t&0&t&t+1&1\\t+1&0&t+1&1&t\\\end{array}After
1^{-1}=1,\quad t^{-1}=t+1,\quad(t+1)^{-1}=t,\quad |F_4|=4Conditions: none
Rule: calc · Uses: G1, S08
Leaf basis: calc
All three nonzero elements of F4 have two-sided inverses, so the nonprime number 4 can be a field order.
Before
F_3\text{ ring laws and inverse list}After
F_3\text{ is a field};\quad D\text{ finite domain}\Rightarrow D\text{ field}Conditions: none
Rule: mko_definition · Uses: G1, S02, S04, S07
Leaf basis: mko_definition
The complete F3 certificate satisfies the field definition; the finite-domain lemma supplies every nonzero inverse through injective-to-surjective reasoning.
No substeps in this guide.
S10 · premise_audit · G1, S02, S04, S07 — The conclusion covers every nonzero element, not merely 2/2, and does not classify arbitrary finite commutative rings.S09 · premise_audit · G1, S08 — t(t+1)=t^2+t=1; 4=2^2 is a prime power and does not refute the prime-power restriction.S08 · premise_audit · G1 — Thus polynomial ring laws descend to these four remainders; the tables are not arbitrary unqualified operations.S07 · premise_audit · G1, S06 — Surjectivity concerns the same finite set D; an infinite-domain injection need not be onto.S06 · premise_audit · G1 — The cancellation uses no assumed inverse; the zero-divisor condition forces x-y=0, avoiding circularity.S05 · premise_audit · G1, S04 — Multiplying back by 2 yields 2; this audit relies on invertibility and does not permit 0/0.S04 · premise_audit · G1, S03 — Check 2*2=1 in both orders; zero must not enter the inverse list.S03 · premise_audit · G1, S02 — Changing representatives by multiples of 3 changes sums and products by multiples of 3, so the laws descend.S02 · premise_audit · G1, S01 — Each inverse must sum to zero; 1+2=0 is specifically modulo 3.S01 · premise_audit · G1 — Each row has three entries and the zero row fixes every element; this is not a sample.Before a/b check b is nonzero; 2 is a valid denominator in F3.
Multiplication by 2 on Z is injective but misses 1, so finiteness cannot be omitted.
reject — 2 is nonzero yet 2*2=0, and no product 2*x is 1.reject — Zero times every element is zero, so it has no multiplicative inverse.reject — The listed F4 tables and inverses give order 4, a prime power but not a prime.F3 is a field and 2/2=1; nonzero multiplication maps supply inverses in a finite domain.
2 declared checks for this guide; only the listed expressions are covered.
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