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The characteristic of a field is zero or a prime number
Statement
The characteristic of a field is either or a prime number.
Facts & Assumptions
Given: A field .
If the set of positive with is nonempty, the characteristic is its least element; otherwise it is (The characteristic of a ring: the least with when one exists, and otherwise).
A natural number greater than is prime exactly when it has no factorization into two natural numbers strictly between and itself (Prime and composite integers: is prime when and its only positive divisors are and ).
A field is an integral domain, so a product of two nonzero elements cannot be zero (Every field is a commutative ring with ; it is an integral domain, and it is a commutative division ring).
Proof
If no positive multiple of is zero, [L1] gives characteristic . Otherwise write . Since , one has .
Suppose, for contradiction, that this positive is not prime. By [L2], write with and .
Minimality of gives and , while their product is . This contradicts [L3].
Therefore the positive characteristic is prime, completing both cases.
Depends on
- The characteristic of a ring: the least $n \ge 1$ with $n \cdot 1_R = 0$ when one exists, and $0$ otherwise
- Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$
- Every field is a commutative ring with $1 \ne 0$; it is an integral domain, and it is a commutative division ring
Used by
- Frobenius linear systems have nonreduced general members Counterexample
- The equation must define the intended scheme Counterexample
- Finite power map of the affine line Example
- In characteristic p the only pᵏ-th root of unity is 1, and t^pᵏ-1=(t-1)^pᵏ Proposition
- A field's prime subfield is isomorphic to ℚ in characteristic zero and to Fₚ in characteristic p Theorem
- For every n≥1 there are infinitely many primes p with p≡1 (mod n) Theorem
- Frobenius x↦ xᵖ is an injective endomorphism in characteristic p, and an automorphism for finite fields Theorem
- K(μₘ)K(μₙ)=K(μ_lcm(m,n)) Theorem
- tⁿ-1 is separable over K exactly when the characteristic does not divide n, and then a splitting field carries n distinct n-th roots of unity Theorem
Dependency tree · two levels
34 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)