Review and study pointers for Exam 3

The test will cover the material in Chap. 8.4 (Geometric Constructions), 8.5 (Finite Fields), 9.3 (Automorphisms of Fields) and 9.3 (Splitting Fields). Notice that by this time we have a great deal of  hierarchically interdependent subject matter, such as irr(a, F), algebraic extensions, evaluation homomorhisms, kernels, etc., etc. You should be sure to have a firm grasp of the theorems and concepts that lead up to our present discussion of field extensions. As always, there are three basic areas: 

• Definitions (approximately 25%) 
• Computations and examples (approximately 40%)
• Proofs of Theorems or Lemmas covered in class or other simple facts (approximately 35%)

Definitions

• Constructible number
• GF(pⁿ)
• Primitive root of unity
• Conjugate elements
• Automorphism
• Fixed field of a set of automorphisms
• G(E/F), (Galois) group of E over F
• Splitting field
• A polynomials splits in E
• Separable polynomial, element, or extension.

Computations and examples

Be able to give examples of :

• constructible or non-constructible numbers,
• finite fields,
• primitive roots of unity,
• fixed fields,
• conjugate elements,
• compute some Galois  groups  polynomials over the rationals, (e.g. Q(Ï 2, Ï 3 ) over Q),
• compute some splitting fields and Galois  groups  polynomials over the rationals, (e.g. f(x) = x³-2  over Q).

Proofs

Be able to reproduce the proofs of: 

• Theorem 8.4.1 (the sum, product and quotient of constructible numbers is constructible)
• Theorems 8.4.9, 10, 11  (impossibility of duplicating the cube, squaring the circle and trisecting an angle by straight-edge and compass)
• Theorems 8.5.1 and 8.5.3 (characterization of finite fields)
• Theorem 9.1.3 (the conjugation isomorphism) it is sufficient to outline the maps involved by understanding and explaining the diagram
• Theorem 8.1.3 (Kroenecker's theorem). Although I will not ask you to reproduce the entire proof, you need to understand the process of constructing an extension field containing a zero of an irreducible polynomial by factoring out the principal ideal generated by it. 
• Theorem 9.1.11 and 9.1.15 (F is a subfield of E_G(E/F))
• Theorem 8.3.3 (If [E:F] is finite then E is an algebraic extension of F)

In addition, I may ask you to give proofs of various simple facts that you may or may not have seen in class.
 

Good Luck. If you have questions ask in class, see me or send me e-mail at ebarbut@uidaho.edu.