TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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A field's prime subfield is isomorphic to $Q$ in characteristic zero and to $F_{p}$ in characteristic $p$

Statement

Let $F_{0}$ be the prime subfield of a field $F$ .

If $char F = p > 0$ , then $F_{0}$ is isomorphic to $F_{p} = Z / p$ .
If $char F = 0$ , then $F_{0}$ is isomorphic to $Q$ .

Each isomorphism sends $1$ to $1_{F}$ .

Facts & Assumptions

Given: A field $F$ and its prime subfield $F_{0}$ .

[L1]

The prime subfield is the intersection of all subfields of $F$ and hence the smallest one (The prime subfield as the intersection of all subfields).

[L2]

The characteristic of $F$ is zero or prime (The characteristic of a field is zero or a prime number).

[L3]

For prime $p$ , the quotient $Z / p$ is a field (For every prime $p$ , the two operations on $Z / p$ make it a field).

[L4]

The rational numbers form a field (The rationals form a field).

[L5]

A field homomorphism preserves addition, multiplication and $1$ (Field homomorphism and embedding); it is therefore injective, its kernel being an ideal of a field that does not contain $1$ .

Proof

technique · direct

1.1givenL2L3L5algebra

Suppose $char F = p > 0$ . The map $Z / p \to F$ given by $[n] \mapsto n \cdot 1_{F}$ is well defined, is a field homomorphism, and is injective; its image is a subfield contained in every subfield of $F$ .

1.2givenL2L4L5algebra

Suppose $char F = 0$ . The map $Z \to F$ , $n \mapsto n \cdot 1_{F}$ , is injective. Sending a rational class $a / b$ with $b \neq 0$ to $(a \cdot 1_{F}) (b \cdot 1_{F})^{- 1}$ is well defined and gives an injective field homomorphism $Q \to F$ .

2.1step 1.1L1

By [L1], that image equals $F_{0}$ , proving the first classification.

2.2step 1.2L1

Its image is a subfield and every subfield of $F$ contains all integer multiples of $1_{F}$ and their nonzero quotients. Hence the image is contained in every subfield and equals $F_{0}$ by [L1].

3.1step 2.1step 2.2L2∎

Steps 2.1 and 2.2 exhaust the alternatives in [L2].

Depends on

Used by

Dependency tree · two levels

20 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

A. W. Knapp, Basic Algebra, 2nd ed., Chapter IX, Section 3 (standard reference, not scraped)

A field's prime subfield is isomorphic to QQ in characteristic zero and to FpFp​ in characteristic pp