Review and study pointers for Exam 1

The test will cover the material in Chap. 5.1- 5.6 and 6.1. In addition you will be responsible for some of the material in Chap. 7 including the proof of Gauss's Lemma on primitive polynomials.

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

• Ring
• A unit of a ring (not to be confused with the unity element of a ring)
• Zero divisor
• Integral domain
• Field
• Characteristic of a ring
• Subring
• Zero of a polynomial
• Irreducible polynomial
• Primitive polynomial
• Ideal
• Factor ring
• Ring homomorphism and isomorphism
• Kernel of a homomorphism
• The field of quotients

Computations and examples

• Be able to do computations in modular arithmetic.
• Be familiar with typical examples of and be able to perform computations in rings such as Z_n (integers modulo n), F[x] (polynomial rings), matrix rings, complex numbers, Gaussian integers, etc..
• Be able to produce examples of commutative and noncommutative rings and subrings or ideals satisfying given properties. Many such examples were covered in class.
• Be able to determine that a given polynomial in Z[x] is irreducible in Q[x] using various tests for irreducibility such as the rational root test, mod p test or Eisenstein's criterion.
• Be able to deal with examples involving factor rings, such as Z[x]/<x²+1>.
• Be able to give and verify examples of ring homomorphisms similar to examples and counterexamples that we discussed in class.
• Be able to give examples of subrings and ideals.
• Be able to determine the image and kernel of a given ring homomorphism..

Proofs

Be able to reproduce the proofs of:

• Theorem 5.4.5 (understand the construction of the field of quotients for an integral domain)
• Theorem 5.5.4 (the Evaluation Homomorphism)
• Corollary 5.6.2 , 5.6.3 and 5.6.4 (a finite subgroup of nonzero elements of a field is cyclic)
• Theorem 5.6.10
• Theorem 5.6.15 (Eisenstein Polynomial)
• Theorem 5.6.20
• Corollary 5.6.17
• Theorems 6.1.16  and 6.1.17 (these constitute the Fundamental Homorphism Theorem for rings)
• Lemma 7.1.24 (Gauss's Lemma)

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% )

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