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

## § 6.1 Algebraic elements

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 u1, u2, . . . , un be elements of F. The smallest subfield of F that contains K and u1, u2, . . . , un will be denoted by

K ( u1, u2, . . . , un ).

It is called the extension field of K generated by u1, u2, . . . , un .
If F = K(u) for a single element u in F, then F is said to be a simple extension of K.

Proposition 6.1.6. Let F be an extension field of K, and let u be an element of F.

(a) If u is algebraic over K, then K(u) K[x]/<p(x)>, where p(x) is the minimal polynomial of u over K.

(b) If u is transcendental over K, then K(u) K(x), where K(x) is the quotient field of the integral domain K[x].

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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