HOMEWORK 1

1. Prove that the subring

F = \{a + b\sqrt{2}; a, b ∈ ℚ\} ⊂ ℝ

is a field.

2. Find characteristic of F and the degree [F : ℚ], where F is a field in Problem 1.

3. Define characteristic of an arbitrary commutative ring. Show that any integer can be characteristic of some ring.

4. Show that f = X^{3} +9X +6 is irreducible in ℚ[X]. Let α be a real root of f. Compute (1+α)^{−1} as a linear combination of 1, α, α^{2} with rational coefficients.

5. Show that f = X^{3} + X + 1 is irreducible in F = (ℤ/2ℤ)[X]. Let α be a root of f in some field extension of F. Compute all the powers α^{n}, n ≥ 0, as linear combinations of 1, α, α^{2}.

6. Let F be a finite field. Prove that the characteristic p of F is positive. Prove that |F | = p^{n} for some positive integer n.

7. Determine the minimal polynomial over ℚ for the element 1 + i.

8. Determine the minimal polynomial over ℚ for the element \sqrt{2} + \sqrt{3}.

9. Let X be a variable over a field F, Y = \frac{X^{2}}{X−1}. Find [F (X) : F (Y)].

10. Let K/F be a field extension, α, β ∈ K. Prove that the extension F (α, β)/F (α + β, αβ) is algebraic.