Galois Theory: A Study of Cyclotomic Field ... - Scripps College

4 The Basics of Field Theory
Definition. An element α ∈ K is algebraic over F if α is a root of a nonzero
polynomial f(x) ∈ F [x]. For example, √ 2 ∈ R is algebraic over Q since it is
a root of x 2 − 2.
Proposition 1. Let α be an algebraic element over the field F and f(x) be its
irreducible polynomial of degree n. Then {1, α, . . . , α n−1 } is a basis for F [α] as a
vector space over F .
Proof. See [A, p.495]
The degree of a field extension K/F is the dimension of K as a vector
space over F and is denoted [K : F ].
A reason to study field extensions is to understand roots of polynomials.
The polynomial x 2 + 1, for example, would not have any roots in R.
However, the complex numbers C, would be a field extension of R (and
thus an extension of Q) that would allow us to find roots of x 2 + 1. It is
easy to show that C is a field under normal addition and multiplication, so
every nonzero element in C has an inverse.
We can consider a field K in which the polynomial f(x) has its roots.
We will assume that the polynomial f(x) is irreducible over F [x] since if
there is a root of a factor of f(x), then it would certainly be a root of f(x)
itself.
Let K be a field generated by a single element α over F . That is to say,
every element β in K can be expressed as β = a 0 +a 1 α+a 2 α 2 +· · · a n α n for
some n and some a 0 , a 1 , a 2 , . . . , a n ∈ F. Then we can write K = F (α). We
say K is a simple extension of F and α is a primitive element for the extension.