TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The quotient Z/2 is the binary field

Statement

The quotient operations on $Z / 2 Z$ make the carrier $F_{2}$ of Binary residue carrier and operations a field. This is only the modulus- $2$ specialization of the published prime-modulus theorem; it does not assert a classification of finite fields.

Facts & Assumptions

Given: The binary residue carrier and its quotient operations.

[F1]

Binary residue carrier and operations: $F_{2}$ is the two residue classes $[0]_{2}, [1]_{2}$ with addition and multiplication induced from $Z$ .

[F2]

Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$ : an integer $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$ .

[F3]

For every prime $p$ , the two operations on $Z / p$ make it a field: for every prime $p$ , the quotient operations on $Z / p Z$ make it a field.

Proof

1.1givenF2algebra

The integer $2$ is greater than $1$ . If a positive integer $d$ divides $2$ , write $2 = d q$ with positive integer $q$ . If $d = 1$ it is the first allowed divisor; if $d \neq 1$ , then $d \geq 2$ , which forces $q = 1$ and $d = 2$ . Thus the only positive divisors are $1$ and $2$ , so $2$ is prime by [F2].

2.1givenF1F3step 1.1algebra∎

Apply [F3] with $p = 2$ using step 1.1. The carrier and both operations in [F1] are exactly those of $Z / 2 Z$ , so every field axiom supplied by that specialization holds for $F_{2}$ .

Depends on

Used by

Gaussian elimination over F2 for Simon recovery Algorithm
Simon independent-sample rank process Definition
The coordinate vector space over F2 Definition
The Simon annihilator is an (n−1)-dimensional binary subspace Lemma
F2 row reduction returns the unique Simon mask Theorem
Simon rank growth probability Theorem

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ronald de Wolf, Quantum Computing: Lecture Notes (arXiv:1907.09415v6, 2026) (standard reference, not scraped)
John Preskill, Quantum Information, Chapter 6: Quantum Algorithms (updated November 2020) (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (17 April 2025) (standard reference, not scraped)