# Exercises

### Exercise 1

(For Thursday January 21)
From the book:
11.1: 1, 2, 3, 8
11.3: 2, 4, 8
15.1: 1, 2, 4, 6
Other problems:
(1) Problem posed by Fermat: prove that the Diophantine equation y² + 2 = x³ has only two solutions, namely x = 3, y = ± 5. Hint: use that Z[√-2] is a Euclidean domain with norm given by N(a + b√-2) = a² + 2b².
(2) Let R be a commutative ring. Show that an ideal I ⊆ R is a prime ideal if and only if R/I is an integral domain. Then show that an element p ∈ R is a prime element if and only if the ideal (p) is a prime ideal.
(3) Show that if R is a PID, then every prime element generates a maximal ideal.

### Exercise 2

(For Monday February 8)
From the book:
15.2: 1, 2, 4
15.3: 2, 4, 8, 10
Problems from previous exams:
June 2015: 1, 3
June 2014: 1, 3, 6, 7
August 2013: 1
May 2013: 3, 4

### Exercise 3

From the book:
16.1: 3, 4, 5, 8
16.2: 2, 3, 4, 6
16.4: 1, 3, 7
16.5: 1, 2, 5
Problems from previous exams:
May 2013: 5, 7

### Exercise 4

Problems from previous exams:
June 2015: all
June 2014: all
Aug 2013: all
May 2013: all
Other problems:
(1) Let p_1,…,p_n be distinct prime numbers. Prove that E = Q(√p_1,…,√p_n) is a Galois extension of Q of degree 2^n.
(2) Show that the Galois group G(E/Q) is isomorphic to (Z/(2))^n.