Math 392 Homework 5 = Midterm review questions SOLUTIONS
1. Dene the function f : Z[i] Z2 by f (x + iy ) = [x + y ] for all x, y Z.
(i) Prove carefully that f is a ring homomorphism.
(ii) What is the kernel of f ?
(iii) Is f any of: onto, 11, isomorphism?
Solutions to nal review sheet
Remember: our nal is at 3.15 on Tuesday March 18!
1. (i) Let R be a ring. What is an irreducible element x R?
(ii) Determine which of the following polynomials in Z2 [x] are irreducible, explaining your method
carefully. For
392 Homework 1 solutions
Exercises 3.2: 1, 2, 3(a)(b)(c),4,6(b)(c)(d),7, 16.
1. Suppose f (x) = xn +an1 xn1 + +a0 C[x] is a monic polynomial
of degree n with roots c1 , . . . , cn . Prove that the sum of the roots is
an1 and the product is (1)n a0 .
Proof
392 Homework 2 solutions
Exercises 3.2: 10. Decide whether f (x) = x3 2 is irreducible in
Q[ 2][x].
The roots are 3 2, 3 2e2i/3 . None of these belong to Q[ 2].
Since it is a cubic no roots implies its irreducible.
Exercises 4.2: 1 (a) Prove that the f
392 Homework 3 solutions
Suppose that f (x) is a monic polynomial in Z[x]. Let Q be a root
of f (x). Show that Z. (Hint. Let = a with GCD(a, b) = 1.
b
Let f (x) = xn + an1 xn1 + + a1 x + a0 . Now substitute in for
x and multiply through by bn1 .)
Solutio
392 Homework 4
Exercises 4.1 4(f), 10, 12, 13(a), 15(a)(e).
4(f) Find all ideals in Q.
We always have the zero ideal (0). Suppose I is a non-zero ideal.
Then it contains a non-zero element, i.e. a unit since Q is a
eld. But once an ideal contains a unit
392 Homework 6
1. Make a table showing how the polynomials (xn 1) factorize into
irreducibles over Q for each n = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Prove
that the factors you nd really ARE irreducible.
n
1
2
3
4
5
6
7
8
9
10
11
12
Factors
(x 1)
(x 1)(x
392 Homework 7 solutions
Exercises 4.1: 6, 18(a)(b)(c)
6 Prove that : Zp Zp , a ap is a ring homomorphism.
Solution. Obviously 1 goes to 1. Also, (ab) = (ab)p = ap bp =
(a)(b). So it is multiplicative. The hard thing is additivity.
(a) + (b) = ap + bp ,
392 Homework 8 solutions
Section 4.3: 1(a)(b), 2(a)(b), 5, 8, 10(a)(b) (When youve done question
10, you should know which out of the Gaussian integers 1 + i and 3 + i is
irreducible in Z[i] . . . ).
1(a) Find GCD(8 + 6i, 5 15i).
Solution. Euclidean algor
Math 392 Homework 9 = Final review questions
This homework will not be graded. We will spend the Wednesday and Friday classes of dead
week mainly on revision then I will hand out solutions to these review questions and go
over any problems.
Also on Wednes