1.2. The characteristic of a field. The map

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Z → F, n → 1F + 1F + ・ ・ ・ + 1F (ntimes),

is a homomorphism of rings.

Case 1: Kernel = (0);the n n ・ 1F = 0 =⇒ n = 0 (in Z). The map Z → F extends to a

homomorphism Q _→ F, m

n

→ (m・ 1F )(n ・ 1F )−1. Thus F contains a copy of Q. In this case,

we say that F has characteristic zero.

Case 2: Kernel _= (0), i.e., n ・ 1F = 0 some n _= 1. The smallest such n will be a

prime p (else F will have nonzero zero-divisors), and p generates the kernel. In this case,

{m ・ 1F | m ∈ Z} ≈ Fp, and F contains a copy of Fp. We say that F has characteristic p.

The fields Fp, p prime, and Q are called the prime fields. Every field contains a copy of

one of them.

Remark 1.3. The binomial theorem

(a + b)m = am +

_

m

1

_

am−1b + ・ ・ ・ +

_

m

r

_

am−rbr + ・ ・ ・ + bm

holds in any ring. If p is prime, then p|

_p

r

_

for all r, 1 ≤ r ≤ p − 1. Therefore, when F has

characteristic p, (a + b)p = ap + bp. Hence a → ap is a homomorphism F → F, called the

Frobenius endomorphism of F. When F is finite, it is an isomorphism, called the Frobenius

automorphism.

2 J.S. MILNE

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