2.10 The Primitive Element Theorem

Back

Let F _ E be a field extension. We say that this extension is simple , or that

E has a primitive element over F if there exists _ 2 E such that E = F(_).

Theorem 2.10.1 Let F _ E be a field extension with [E : F] < 1. Then E

has a primitive element over F if and only if there are only a finite number

of fields between F and E.

Let F _ E be a field extension, and let _ 2 E. We say that _ is separable

over F if its minimal polynomial m_(x) 2 F[x] is separable. If every element

of E is separable over F, then we call E a separable extension of F.

Corollary 2.10.1.1 [Primitive Element Theorem] Let F _ E be a finite

dimensional separable field extension. Then E contains a primitive element

over F.

Exercises 2.8

1. Find a primitive element for Q(

p

2,

p

3) over Q

2. Find a primitive element for a splitting field for x4 − 2 over Q.

3. Let F _ E be a finite Galois extension with Galois group G. If _ 2 E,

prove that

[F(_) : F] = [G : StabG(_)].

4. Let F be any field and let x be an indeterminate over F. Let E = F(x).

Let y be an indeterminate over E, and let K = E[y]/(y2 − x(x − 1)),

regarded as an extension field of E. Show that K is a simple extension

of F (though obviously not of finite dimension).

5. Let Q _ K be a field extension. Assume that whenever _ 2 Q, then p

_ 2 K. Prove that [K : Q] = 1. (Compare with Exercise 8 of

Section 2.1.)

6. Let F be a field and let x be indeterminate over F. Are there finitely

or infinitely many subfields between F and F(x)?

2.10. THE PRIMITIVE ELEMENT THEOREM 71

7. Let F _ E be a field extension such that there are finitely many subfields

between F and E. Prove that E is a finite extension of F.

8. Let F _ E be a finite separable extension and assume that E = F(_, _)

for some _, _ 2 E. Prove that E = F(_ +a_) for all but finitely many

a 2 F.

Chapter 3 Elementary Factorization Theory

Навигация