Fields - Elliptic Curves: Number Theory and Cryptography - page 480

2. φ q is an automorphism of F q .Inparticular,

φ q ( x + y )= φ q ( x )+ φ q ( y ) , q ( xy )= φ q ( x ) φ q ( y )

for all x, y ∈ F q .

3. Let α ∈ F q .Then

φ q ( α )= α. α ∈ F q n ⇐⇒

PROOF L and

every element of L is algebraic over K ,then L = K . This can be proved
as follows. If α is algebraic over L and L is algebraic over K , then a ba sic
property of algebraicity is that α is then algebraic over K . Therefore, L is
algebraic over K and is algebraically closed.

Part (1) is a special case of a mor e gen er al fact: If K ⊆

Therefore, it is an algebraic

closure of K .
Part (3) is just a restatement of Theorem C.1, with q n in place of q .

We now prove part (2). If 1 ≤ j ≤ p − 1, the binomial coe cient j has a
factor of p in its numerator that is not canceled by the denominator, so
p
j

≡ 0(mod p ) .

Therefore,

( x + y ) p = x p + p
1

x p− 1 y + p
2

x p− 2 y 2 + ··· + y p

= x p + y p

since we are working in characteristic p . An easy induction yields that

( x + y ) p n = x p n + y p n

for all x, y ∈ F p . Thisimpliesthat φ q ( x + y )= φ q ( x )+ φ q ( y ). The fact
that φ q ( xy )= φ q ( x ) φ q ( y )isclear. Thisprovesthat φ q is a homomorphism
of fields. Since a homomorphism of fields is automatically injective (see the
discussi on preceding Proposition C.5), it remains to prove that φ q is surjective.
If α ∈ F p ,then α ∈ F q n for some n ,so φ q ( α )= α . Therefore, α is in the
image of φ q ,so φ q is surjective. Therefore, φ q is an automorphism.

In Appendix A, it was pointed out that F p = Z p is a cyclic group, gener-
ated by a primitive root. More generally, it can be shown that F q is a cyclic
group. A useful consequence is the following.

PROPOSITION C.3
Let m be a positive integer with p m and let μ m be the group of mth roots of
unity. Then

μ m ⊆ F q ⇐⇒ m|q − 1 .