Spring 2007 – Mathematics 4321 – Review Test 3

— Dr. Cordero

I. Define the following terms:

1. degree of an extension field
2. algebraic elements; algebraic extension
3. transcendental element
4. simple extension; simple algebraic extension
5. irreducible polynomial for  over F
6. splitting field of a polynomial
7. prime subfield
8. automorphism
9. F-automorphism of K, Galois group of K over F

II. Study the following theorems:

1. Kronecker’s Theorem.
2. Theorem 42.1, 42.2, 42.3, 42.4, 43.1, 43.2, 43.3, 44.1, 44.2, 44.3, 44.4, 44.5, 45.1,  45.2, Freshman’s dream (Lemma 45.1), 45.3, 46.2

    3. Theorem: If  is algebraic over F, then there exist a monic irreducible polynomial  of degree n such that . Every element  is of the form  for .

    3. Theorem: A finite field  of order  exists for every prime power .

    4. Theorem: Let K be an extension of F and . If  is a root of f(x) and , then is also a root of f(x).

    5. Theorem: Let K=F() be an algebraic extension of F. If  with  for every , then .

    6. Theorem: If K is the splitting field of a polynomial f(x) of degree n in F[x], then Gal(K/F) is isomorphic to  a subgroup of .

III. Do all assigned problems from Sections 42-46.