Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed. © 1996

## § 4.4 Existence of Roots

Definition 4.4.1. Let E and F be fields. If F is a subset of E and has the operations of addition and multiplication induced by E, then F is called a subfield of E, and E is called an extension field of F.

Definition 4.4.2. Let F be a field, and let p(x) be a fixed polynomial over F. If a(x), b(x) belong to F[x], then we say that a(x) and b(x) are congruent modulo p(x), if p(x) | (a(x)-b(x)), written

a(x) b(x) (mod p(x)) .

The set

{ b(x) in F[x] | a(x) b(x) (mod p(x)) }

is called the congruence class of a(x), and will be denoted by [a(x)].
The set of all congruence classes modulo p(x) will be denoted by

F[x] / <p(x)> .

Proposition 4.4.3. Let F be a field, and let p(x) be a nonzero polynomial in F[x]. For any polynomial a(x) in F[x], the congruence class [a(x)] modulo p(x) contains a unique representative r(x) with deg(r(x))<deg(p(x)) or r(x)=0.

Proposition 4.4.4. Let F be a field, and let p(x) be a nonzero polynomial in F[x]. For any polynomials a(x),b(x),c(x), and d(x) in F[x], the following conditions hold:

(a) If a(x) c(x) (mod p(x)) and b(x) d(x) (mod p(x)), then

a(x)+b(x) c(x)+d(x) (mod p(x))

and

a(x)b(x) c(x)d(x) (mod p(x)).

(b) If gcd(a(x),p(x))=1, then

a(x)b(x) a(x)c(x) (mod p(x))

implies

b(x) c(x) (mod p(x)).

Proposition 4.4.5. Let F be a field, and let p(x) be a nonzero polynomial in F[x]. For any polynomial a(x) in> F[x], the congruence class [a(x)] has a multiplicative inverse in F[x]/<p(x)> if and only if gcd(a(x),p(x))=1.

Theorem 4.4.6. Let F be a field, and let p(x) be a nonconstant polynomial over F. Then F[x]/<p(x)> is a field if and only if p(x) is irreducible over F.

Definition 4.4.7. Let F1 and F2 be fields. A function : F1 -> F2 is called an isomorphism of fields if

(i) is one-to-one and onto,

(ii) (a+b) = (a) + (b),   for all a,b in F1, and

(iii) (ab) = (a) (b),   for all a,b in F1.

Theorem 4.4.8. [Kronecker] Let F be a field, and let f(x) be any nonconstant polynomial in F[x]. Then there exists an extension field E of F and an element u in E such that f(u)=0.

Corollary 4.4.9. Let F be a field, and let f(x) be any nonconstant polynomial in F[x]. Then there exists an extension field E of F over which f(x) can be factored into a product of linear factors.

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