Fields - Elliptic Curves: Number Theory and Cryptography - page 478

Appendix C

Fields

Let K be a field. There is a ring homomorphism ψ : Z → K that sends
1 ∈ Z to 1 ∈ K .If ψ is injective, then we say that K has characteristic 0.
Otherwise, there is a smallest positive integer p such that ψ ( p )=0. Inthis
case, we say that K has characteristic p .If p factors as ab with 1 <a≤
b<p ,then ψ ( a ) ψ ( b )= ψ ( p )=0,so ψ ( a )=0or ψ ( b ) = 0, contradicting the
minimality of p . Therefore, p is prime.
When K has characteristic 0, the field Q of rational numbers is contained
in K .When K has characteristic p , the field F p of integers mod p is contained
in K .
Let K and L be fields with K ⊆ L .If α ∈ L ,wesaythat α is algebraic
over K if there exists a nonconstant polynomial

f ( X )= X n + a n− 1 X n− 1 + ··· + a 0

with a 0 ,...,a n− 1 ∈ K such that f ( α )=0. Wesaythat L is an algebraic

over K ,orthat L is an algebraic extension of K , if every el em ent of L is
algebraic over K .An algebraic closure of a field K is a field K containing
K such that

1. K is algebraic over K .

2. Ev ery nonconstant pol ynomial g ( X ) with coe cients in K has a root in
K (this means that K is algebraically closed).

If g ( X ) has degree n and has a root α ∈ K ,thenwecanwrite g ( X )=
( X 1. By induction, we see that g ( X )

has exactly n roots (counting multiplicity) in K .
It can be shown that every field K has an algebraic closure, and that any two
algebraic closures of K are isomorphic. Throughout the topic, we implicitly
assume that a particular algebraic closure of a field K has been chosen, and
we refer to it as the algebraic closure o f K .
When K = Q , the algebraic closure Q is the set of complex numbers that
are algebraic over Q .When K = C , the algebraic closure is C itself, since
the fundamental theorem of algebra states that C is algebraically closed.

− α ) g 1 ( X )with g 1 ( X )ofdegree n −