Math 361, Spring 2013, Assignment 2

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different.

- Goethe

Carefully define the following terms, then give one example and one non-example of each:

1. Polynomial with coefficients in the ring \(R\).
2. Evaluation homomorphism.
3. Root (or zero) of a polynomial.
4. Irreducible polynomial (over a field \(F\)).

Carefully state the following theorems (you need not prove them):

1. Theorem on division in \(F[x]\) (Theorem 23.1).
2. Factor theorem.
3. Theorem on finite subgroups of the multiplicative group of a field (Corollary 23.6).
4. Theorem on reducibility of quadratic and cubic polynomials. (Also give an example to show that this result is false for polynomials of degree higher than three.)
5. Eisenstein's Criterion.
6. Theorem on unique factorization in \(F[x]\).

Solve the following problems:

• Section 22, problems 3, 5, 7, 11, 13, 23, and 27.
• Section 23, problems 1, 5, 9, 15, 19, 21, 25, and 29.

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