122 The Idea Behind Galois Theory
Definition 8.16. Let R : K be a radical extension. The height of R : K is the smallest
integer h such that there exist elements α
1
,...,α
h
∈ R and primes p
1
,..., p
h
such
that R = K(α
1
,...,α
h
) and
α
p
j
j
∈ K(α
1
,...,α
j−1
) 1 ≤ j ≤ h
where when j = 1 we interpret K(α
1
,...,α
j−1
) as K.
Proposition 8.9 shows that the height of every radical extension is defined.
We prove Theorem 8.15 by induction on the height of a radical extension R that
contains x. The key step is extensions of height 1, and this is where all the work is
put in.
Lemma 8.17. Let M be a subfield of L such that K ⊆ M, and let a ∈ M, where a is
not a pth power in M. Then
(1) a
k
is not a pth power in M for k = 1, 2, . . . , p −1.
(2) The polynomial m(t) = t
p
−a is irreducible ...