Exercises VII

1. Construct subfields of C which are splitting fields of the following polynomial over Q:

X^{3} − 1, X^{4} + 5X^{2} + 2, X^{6} − 8.

What are their degrees over Q?

2. Construct the splitting fields over F_{3} for the following polynomials:

X^{3} + 2X + 1, X^{3} + X^{2} + X + 2.

Are these fields isomorphic?

3. Which of the following extensions are normal?

Q(X)/Q, Q(\sqrt{−5})/Q, Q(α)/Q, Q(α, \sqrt{5})/Q(α).

Here α is a real seventh root of 5.

4. Construct the normal closures of the following extensions:

Q(α)/Q, Q(β)/Q, Q(\sqrt{2}, \sqrt{3})/Q, Q(γ, \sqrt{2})/Q, Q(δ)/Q.

Here α^{5} = 3, β^{7} = 2, γ^{3} = 2 are real solutions and δ is a root of X^{3} − 3X^{2} + 3.

5. Find the Galois groups over Q of the extensions you constructed in exercise 4.