16.4. VECTOR SPACES AND FIELDS∗ 393

You can do it with partial fractions as above.

1(x2 +1)(ax+b)

=b−ax

(a2 +b2)(x2 +1)+

a2

(a2 +b2)(ax+b)

and so

1 =1

a2 +b2 (b−ax)(ax+b)+a2

(a2 +b2)

(x2 +1

)Thus

1a2 +b2 (b−ax)(ax+b)∼ 1

and so

[ax+b]−1 =[(b−ax)]a2 +b2 =

b−a [x]a2 +b2

You might find it interesting to recall that (ai+b)−1 = b−aia2+b2 .

16.4.3 The Algebraic NumbersEach polynomial having coefficients in a field F has a splitting field. Consider the caseof all polynomials p(x) having coefficients in a field F⊆G and consider all roots whichare also in G. The theory of vector spaces is very useful in the study of these algebraicnumbers. Here is a definition.

Definition 16.4.27 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.

Theorem 16.4.28 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 the 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 and irreducible such that q(a) =0.

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)is irreducible, l (x) = 1 showing that p(x) = q(x). ■

Definition 16.4.29 For a an algebraic number, let deg(a) denote the degree of the minimalpolynomial of a.

Also, here is another definition.

Definition 16.4.30 Let a1, · · · ,am be in A. A polynomial in {a1, · · · ,am} will be an expres-sion of the form

∑k1···kn

ak1···knak11 · · ·a

knn

where the ak1···kn are in F, each k j is a nonnegative integer, and all but finitely many of theak1···kn equal zero. The collection of such polynomials will be denoted by

F [a1, · · · ,am] .