7.3. LOTS OF FIELDS 205

7.3.3 The Algebraic Numbers

Each polynomial having coefficients in a field F has a splitting field. Consider the case of allpolynomials p (x) having coefficients in a field F ⊆ G and consider all roots which are alsoin G. The theory of vector spaces is very useful in the study of these algebraic numbers.Here is a definition.

Definition 7.3.27 The algebraic numbers A are those numbers which are in G and alsoroots of some polynomial p (x) having coefficients in F. The minimal polynomial of a ∈ Ais defined to be the monic polynomial p (x) having smallest degree such that p (a) = 0.

The next theorem is on the uniqueness of the minimal polynomial.

Theorem 7.3.28 Let a ∈ A. Then there exists a unique monic irreducible polynomial p (x)having coefficients in F such that p (a) = 0. This polynomial is the minimal polynomial.

Proof: Let p (x) be a monic polynomial having smallest degree such that p (a) = 0.Then p (x) is irreducible because if not, there would exist a polynomial having smallerdegree which has a as a root. Now suppose q (x) is monic with smallest degree such thatq (a) = 0. Then

q (x) = p (x) l (x) + r (x)

where if r (x) ̸= 0, then it has smaller degree than p (x). But in this case, the equationimplies r (a) = 0 which contradicts the choice of p (x). Hence r (x) = 0 and so, since q (x)has smallest degree, l (x) = 1 showing that p (x) = q (x). ■

Definition 7.3.29 For a an algebraic number, let deg (a) denote the degree of the minimalpolynomial of a.

Also, here is another definition.

Definition 7.3.30 Let a1, · · · , am be in A. A polynomial in {a1, · · · , am} will be an ex-pression of the form ∑

k1···kn

ak1···knak11 · · · akn

n

where the ak1···knare in F, each kj is a nonnegative integer, and all but finitely many of the

ak1···knequal zero. The collection of such polynomials will be denoted by

F [a1, · · · , am] .

Now notice that for a an algebraic number, F [a] is a vector space with field of scalars F.Similarly, for {a1, · · · , am} algebraic numbers, F [a1, · · · , am] is a vector space with field ofscalars F. The following fundamental proposition is important.

Proposition 7.3.31 Let {a1, · · · , am} be algebraic numbers. Then

dimF [a1, · · · , am] ≤m∏j=1

deg (aj)

and for an algebraic number a,dimF [a] = deg (a)

Every element of F [a1, · · · , am] is in A and F [a1, · · · , am] is a field.