# 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.