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**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 theThe 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)), thena(x)+b(x) c(x)+d(x) (mod p(x))

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

**(b)**If gcd(a(x),p(x))=1, thena(x)b(x) a(x)c(x) (mod p(x))

impliesb(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 F_{1} and F_{2} be fields. A function
: F_{1} -> F_{2}
is called an ** isomorphism of fields** if

**(i)**is one-to-one and onto,**(ii)**(a+b) = (a) + (b), for all a,b in F_{1}, and- (iii) (ab) = (a) (b), for all a,b in F
_{1}.- (iii) (ab) = (a) (b), for all a,b in F

**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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