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Definition 6.1.1.
The field F is said to be an
extension field
of the field K if K is a subset of F which
is a field under the operations of F.
Definition 6.1.2.
Let F be an extension field of K and let u be an element of F.
If there exists a nonzero polynomial f(x) in K[x] such that f(u)=0,
then u is said to be
algebraic
over K. If there does not exist such a polynomial, then u is said to be
transcendental
over K.
Proposition 6.1.3.
Let F be an extension field of K,
and let u be an element of F that is algebraic over K.
Then there exists a unique monic irreducible polynomial
p(x) in K[x] such that p(u)=0.
It is characterized as the monic polynomial of minimal degree
that has u as a root.
Furthermore, if f(x) is any polynomial in K[x] with f(u)=0, then p(x) | f(x).
Definition 6.1.4.
Let F be an extension field of K,
and let u be an algebraic element of F.
The monic polynomial p(x) of minimal degree in K[x] such
that p(u)=0 is called the
minimal polynomial
of u over K. The degree of the minimal polynomial of u over K is called the
degree
of u over K.
Definition 6.1.5.
Let F be an extension field of K, and let
u_{1},
u_{2},
. . . ,
u_{n} be elements of F.
The smallest subfield of F that contains K and
u_{1},
u_{2},
. . . ,
u_{n}
will be denoted by
K ( u_{1}, u_{2}, . . . , u_{n} ).
It is called the extension field of K generated by u_{1}, u_{2}, . . . , u_{n} .
Proposition 6.1.6.
Let F be an extension field of K, and let u be an element of F.
The next proposition is simply a restatement of
Kronecker's theorem.
Proposition 6.1.7.
Let K be a field and let p(x) be any irreducible polynomial in K[x].
Then there exists an extension field F of K and an element u in F
such that the minimal polynomial of u over K is p(x).
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