Review and study pointers for Exam 2
The test will cover the material in Chap. 6.1, 6.2, 8.1,
8.2 and 8.3. There will be no questions on 8.2 beyond the definitions of linear
combinations, linear independence, basis and dimension in vector spaces. Most of
the material in 6.1 was covered in the first exam, but I am including it because
the concepts of homomorphism and factor rings are central to the subsequent
material and should be included in any review study in preparation for the test.
Only the first half of 8.3 (algebraic extensions) will be covered. The material
on algebraic closure will not be included.
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
- Homomorphism, kernel and factor ring
- Maximal ideal
- Prime ideal
- Prime field
- Principal ideal
- Extension field
- Algebraic and transcendental element
- irr(a, F)
and deg(a, F)
- Simple extension
- Span, linear combination, basis and dimension of a
vector space.
- Algebraic extension
- Finite extension
Computations and examples
Be able to:
- give examples of prime ideals and maximal ideals,
- determine irr(a, F)
and deg(a, F)
for a given a in an extension
E of a field F,
- deal with examples involving factor rings, such as Z[x]/<x2+1>,
- construct finite fields with pn elements (e.g. 8 or 27
elements) by factoring out the principal ideal generated by an irreducible
polynomial of degree n over Zp,
- determine whether a given extension is algebraic over a
field F and the degree of that extension,
- find a basis for a finite dimensional extension of a
given field, e.g. Q(Ï
2,
Ï
3 ) over Q,
- compute simple finite extensions, e.g. Q(Ï
2 +
Ï
3 ) = Q(Ï
2,
Ï
3 ) or Q(21/3,
21/2) = Q(21/6) using dimension arguments.
Proofs
Be able to reproduce the proofs of:
- Theorem 6.2.9 (M is a maximal ideal of R
iff R/M is a field)
- Theorem 6.2.15 (N is
a prime ideal of R iff R/N is an integral domain)
- Theorem 6.2.24 (if F is a field then every ideal
of F[x] is principal; use the division algorithm for
polynomials)
- Theorem 6.2.25 (<p(x)> is a maximal ideal
of F[x] iff p(x) is irreducible)
- 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 8.1.13 (irr(a,
F) divides any other f(x) for which f(a)=0;
use the division algorithm for polynomials)
- Theorem 8.3.3 (If [E:F] is finite then
E is an algebraic extension of F)
- Theorem 8.3.4 ([K:F]=[K:E][E:F])
In addition, I may ask you to give proofs of various simple facts that you
may or may not have seen in class (about 10% or perhaps extra credit)
Good Luck. If you have questions ask in class, see me or send me e-mail at
ebarbut@uidaho.edu.
