3.4. POLYNOMIALS AND FIELDS 69

and so 1 = (−x)(x2 + x+1

)+(x+1)

(x2 +1

)which implies 1∼ (−x)

(x2 + x+1

)and so

the inverse is [−x] .The following proposition is interesting. It was essentially proved above but to empha-

size it, here it is again.

Proposition 3.4.16 Suppose p(x) ∈ F [x] is irreducible and has degree n. Then every ele-ment of G = F [x]/(p(x)) is of the form [0] or [r (x)] where the degree of r (x) is less thann.

Proof: This follows right away from the Euclidean algorithm for polynomials. If k (x)has degree larger than n− 1, then k (x) = q(x) p(x)+ r (x) where r (x) is either equal to 0or has degree less than n. Hence [k (x)] = [r (x)] . ■

Example 3.4.17 In the situation of the above example where the polynomial is x2 + 1 ir-reducible in R(x), find [ax+b]−1 assuming a2 + b2 ̸= 0. Note this includes all cases ofinterest thanks to the above proposition.

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 1

a2+b2 (b−ax)(ax+b)∼ 1and 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 . Didn’t this just produce

the complex numbers algebraically? If, instead of R you used Q this would have justproduced a field Q+ iQ.

3.4.1 The Algebraic Numbers and Minimum PolynomialEach 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 3.4.18 Let F and G be two fields, F ⊆ G. The algebraic numbers A are thosenumbers which are in G and also roots of some polynomial p(x) having coefficients in F.The minimum polynomial1 of a ∈ A is defined to be the monic polynomial p(x) havingsmallest degree such that p(a) = 0. It is also often called the minimum polynomial.

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

1I grew up calling this and similar things the minimal polynomial, but I think it is better to call it the minimumpolynomial because it is unique. If you see minimal polynomial, this is what it is.