Extension Field and Polynomials

• (Fraleigh 29.3) Kronecker’s Theorem Let be a field and be a nonconstant polynomial in . Then there exists an extension field of and an such that

  • Proof: Consider an irreducible polynomial that divides . The extension field is the factor ring , which is a field because is a maximal ideal by Fraleigh 27.25. Clearly since we can take the additive coset as our representation for so is indeed an extension field.

    Finally, we can find . The evaluation homomorphism evaluated at can be computed using as a representative of the additive coset which gives us

    Which is precisely .

• An element of an extension field of a field is algebraic over if for some nonzero .

  Otherwise, is transcendental over .

  In other words, is algebraic over if it is the zero of some polynomial over , where .

  • From a structural viewpoint, a transcendental element over behaves as though it were an indeterminate over .

• (Fraleigh 29.12) Let and let . Let be the evaluation homomorphism of into such that for . and .

  Then is transcendental over if and only if gives an isomorphism of with a subdomain of (i.e., if is a one-to-one map)

  • Proof: is transcendental over if and only if for all nonzero if and only if for all nonzero if and only if if and only if is a one-to-one map.

• (Fraleigh 29.13) Let and , where is algebraic over . Then there is an irreducible polynomial such that . This polynomial is uniquely determined up to a constant factor in and is a polynomial of minimal degree in having as a zero.

  If for with , then divides .

  • Another way to say the theorem is that: Every algebraic element has a corresponding unique monic polynomial in [x].
  • Intuition: is a principal ideal generated by some such that is a zero of this polynomial. Any with as a zero must be in . If is of minimal degree, then any other polynomial of the same degree must be a constant multiple of .
  • The unique monic polynomial is called the irreducible polynomial for over denoted . The degree of is the degree of over denoted . That is

• An extension field of is a simple extension of if for some

  • If is algebraic over , then the subfield is the smallest subfield containing and . We find it as follows

    The RHS follows because which is a maximal ideal of .

  • If is transcendental over , then is an Integral Domain denoted . We then define using the Quotient Field

  • In either case, an element in can be expressed as a quotient of polynomials in .

  • Also is the smallest field in containing both and

  • (Fraleigh e29.29) Let and . If is transcendental over but algebraic over , then is algebraic over .

  • (Fraleigh e29.33) Let and be transcendental over . Every element in not in is also transcendental over

  • We can generalize to more than one element. Let . Then the smallest extension field of containing all is denoted

• (Fraleigh 29.18) Let be algebraic over . Let the degree of . Then , we can uniquely express in the form

  Where

• (Fraleigh 30.23) Let be an extension field of and be algebraic over . If then is an -dimensional Vector Space over with basis

  Also, every element is algebraic over and

Topics §

• Algebraic Extension
• Algebraic Closure

Links §

• Fraleigh