2.1 Basics

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We assume that the reader is familiar with the definition of a field; typically

in these notes a field will be denoted in bold face notation: F,K,E, and the

like. The reader should also be familiar with the concept of the characteristic

of a field.

If F and K are fields with F _ K, we say that K is an extension of

F. Of fundamental importance here is the observation that if F _ K is an

extension of fields, then K can be regarded as a vector space over F. It

is customary to call the F-dimension of K the degree of K over F, and to

denote this degree by [K : F]. The following simple result is fundamental.

Proposition 2.1.1 Let F _ E _ K be an extension of fields. Then [K :

F] < 1 if and only if each of [K : E], [E : F] < 1, in which case

[K : F] = [K : E] · [E : F].

If F _ K is a field extension, and if _ 2 K, we write F(_) for the smallest

subfield of K containing F and _. Similarly, we write F[_] for the smallest

subring of K containing both F and _. Clearly,

F(_) =

_

f(_)

g(_) | f(x), g(x) 2 F[x], g(_) 6= 0

_

,

F[_] = {f(_)| f(x) 2 F[x]}.

We say that _ is algebraic over F if there is a non-zero polynomial f(x) 2 F[x]

such that f(_) = 0. When F = Q, the field of rational numbers, and _ is

42

2.1. BASICS 43

algebraic over Q, we say that _ is an algebraic number. If _ is algebraic over

F, then there is a unique monic polynomial of least degree in F[x], called

the minimal polynomial of _, and denoted m_(x), such that m_(_) = 0.

Clearly m_(x) is irreducible in F[x]. If degm_(x) = n, we say that _ has

degree n over F.

The following is frequently useful.

Lemma 2.1.2 Let F _ K, and let _ 2 K. Then _ is algebraic over F if and

only if F(_) = F[_].

Proposition 2.1.3 Let F _ K be a field extension, and let _ 2 K be

algebraic over F, with minimal polynomial m_(x) of degree n.

(a) The map x 7! _ of F[x] ! K induces an isomorphism F[x]/(m_(x)) _=

F(_).

(b) F(_) = F[_] = {f(_)| f(x) 2 F[x], and deg f(x) < n}.

(c) [F(_) : F] = n.

(d) {1, _, . . . , _n−1} is an F-basis for F(_).

In general, if F _ K is a field extension, and if K = F(_), for some

_ 2 K, we say that K is a simple field extension of F. Thus, a very trivial

example is that of C _ R; since C = R(i), we see that C is a simple field

extension of R. We shall see in Section 2.10 that any finite extension of a

field of characteristic 0 is a simple extension (this is the so-called Primitive

Element Theorem).

The result of the above proposition can be reversed, as follows. Let F be a

field, and let f(x) 2 F[x] be an irreducible polynomial. Set K = F[x]/(f(x))

(which is a field since f(x) is irreducible), and regard F as a subfield of K

via the injection F ! K, a 7! a + (f(x)), a 2 F.

Proposition 2.1.4 Let F,K be as above, and set _ = x + (f(x)) 2 K.

Then _ is a root of f(x), and [K : F] = deg f(x).

The point of the above proposition is, of course, that given any field F,

and any polynomial f(x) 2 F[x], we can find a field extension of F in which

f(x) has a root.

By repeated application of Proposition 2.1.4, we see that if f(x) 2 F[x]

is any polynomial, then there is a field K _ F such that f(x) splits completely

into linear factors in K. By definition, a splitting field over F for

44 CHAPTER 2. FIELD AND GALOIS THEORY

the polynomial f(x) 2 F[x] is a field extension of F which is minimal with

respect to such a splitting. Thus it is clear that splitting fields exist; indeed,

if K _ F is such that f(x) splits completely in K[x], and in _1, _2, . . . , _k

are the distinct roots of f(x) in K, then F(_1, _2, . . . , _k) _ K is a splitting

field for f(x) over F. In particular, we see that the degree of a splitting field

for f(x) over F has degree at most n! over F, where n = deg f(x). In the

next section we will investigate the uniqueness of splitting fields.

The next result is easy.

Proposition 2.1.5 If F is a field, and if f(x) is a polynomial F[x] of degree

n, then f(x) can have at most n distinct roots in F.

From the above, one can immediately deduce the following interesting

consequence.

Corollary 2.1.5.1 Let Z _ F× be a finite subgroup of the multiplicative

group of the field F. Then Z is cyclic.

Exercises 2.1

1. Compute the minimal polynomials over Q of the following complex

numbers.

(a)

p

2 +

p

3.

(b)

p

2 + _, where _ = e2_i/3.

2. Let F _ K be a field extension with [K : F] odd. If _ 2 K, prove that

F(_2) = F(_).

3. Assume that _ = a + bi 2 C is algebraic over Q, where a is rational

and b is real. Prove that m_(x) has even degree.

4. Let K = Q( 3

p

2,

p

2) _ C. Compute [K : Q].

5. Let K = Q( 4

p

2, i) _ C. Show that

(a) K contains all roots of x4 − 2 2 Q[x].

(b) Compute [K : Q].

2.1. BASICS 45

6. Let F = C(x), where C is the complex number field and x is an indeterminate.

Assume that F _ K and that K contains an element y such

that y2 = x(x − 1). Prove that there exists an element z 2 F(y) such

that F(y) = C(z), i.e., F(y) is a “simple transcendental extension” of

C.

7. Let F _ K be a field extension. If the subfields of K containing F are

totally ordered by inclusion, prove that K is a simple extension of F.

(Is the converse true?)

8. Let Q _ K be a field extension. Assume that K is closed under taking

square roots, i.e., if _ 2 K, then

p

_ 2 K. Prove that [K : Q] = 1.

(Compare with Exercise 5, Section 2.10.)

9. Let F be a field, contained as a subring of the integral domain R. If

every element of R is algebraic over F, show that R is actually a field.

Give an example of a non-integral domain R containing a field F such

that every element of R is algebraic over F. Obviously, R cannot be a

field.

10. Let F _ K be fields and let f(x), g(x) 2 F[x] with f(x)|g(x) in K[x].

Prove that f(x)|g(x) in F[x].

11. Let F _ K be fields and let f(x), g(x) 2 F[x]. If d(x) is the greatest

common denominator of f(x) and g(x) in F[x], prove that d(x) is the

greatest common denominator of f(x) and g(x) in K[x].

12. Let F _ E1,E2 _ E be fields. Define E1E2 _ E to be the smallest

field containing both E1 and E2. E1E2 is called the composite (or

compositum) of the fields E1 and E2. Prove that if [E : F] < 1, then

[E1E2 : F] _ [E1 : F] · [E2 : F].

13. Given a complex number _ it can be quite difficult to determine

whether _ is algebraic or transcendental. It was known already in

the nineteenth century that _ and e are transcendental, but the fact

that such numbers as e_ and 2

p

2 are transcendental is more recent,

and follows from the following deep theorem of Gelfond and Schneider:

Let _ and _ be algebraic numbers. If

_ =

log _

log _

46 CHAPTER 2. FIELD AND GALOIS THEORY

is irrational, then _ is transcendental. (See E. Hille, American Mathematical

Monthly, vol. 49(1042), pp. 654-661.) Using this result, prove

that 2

p

2 and e_ are both transcendental. (For 2

p

2, set _ = 2

p

2, _ =

2.)

2.2. SPLITTING FIELDS AND ALGEBRAIC CLOSURE 47

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