Fields - Elliptic Curves: Number Theory and Cryptography - page 479

Finite fields

Let p be a prime. The integers mod p form a field F p with p elements. It can
be shown that the number of elements in a finite field is a power of a prime,
and for each power p n of a prime p , there is a unique (up to isomorphism) field
with p n elements. ( Note: The ring Z p n is not a field when n ≥ 2since p does
not have a multiplicative inverse; in fact, p is a zero divisor since p · p n− 1 ≡ 0

(mod p n ).) In this topic, the field with p n elements is denoted F p n .Another
notation that appears often in the literature is GF ( p n ). It can be shown that

F p m ⊆ F p n ⇐⇒ m | n.

The algebraic closure of F p can be shown to be
F p =
n

F p n .

≥ 1

TH EO REM C.1
Let F p be the algebraic closure of F p and let q = p n .Then

F q = {α ∈ F p | α q = α}.

PROOF The group F q of nonzero elements of F q forms a group of order
q − 1, so α q− 1 =1when0= α ∈ F q . Therefore, α q = α for all α ∈ F q .
Recall that a polynomial g ( X ) has multiple roots if and only if g ( X )and
g ( X ) have a common root. Since

d

dX ( X q

X )= qX q− 1

− − 1= − 1

(since q = p n =0in F p ), the po ly nomial X q − X has no multiple roots.

Therefore, there are q distinct α ∈ F p such that α q = α .
Since both sets in the statement of the theorem have q elements and one is
contained in the other, they are equal.

Define the q -th power Frobenius automorphism φ q of F q by the formula

φ q ( x )= x q for all x ∈ F q .

PROPOSITION C.2
Let q be a power of the prime p.

1. F q = F p .