MA4502 Algebraic Number Theory: Problem sheet 2 Please hand in solutions by Monday 6th March 2000

The estimated difficulty (from 1 to 5) is given at the end of each problem 1. (Stickelberger) Show that the discriminant of a number field K satisfies Δ_{K} ≡ 0 or 1 (mod 4). [4]

2. Let p be a prime, and let m(X) ∈ Z[ X] be a monic polynomial of degree d satisfying Eisenstein’s criterion with respect to p (i.e., all non-leading cofficients are divisible by p but the constant coefficient is not divisible by p^{2}). Let α be a root of m(X) = 0, and set K = Q(α). Suppose that p divides |O_{K} : Z[ α]| so that there is β ∈ O_{K} with pβ ∈ Z[ α] but β ∉ Z[ α]. Show that α^{d} is divisible by p in O_{K}, and deduce that for some integer t with 0 ≤ t < d, γ = α^{t}β ∉ Z[ α] but αγ ∈ Z[ α]. Deduce that c/α ∈ O_{K} where c is some integer not divisible by p, and by taking norms derive a contradiction. [3]

3. Suppose that K = Q(α) where α is a zero of the irreducible polynomial m(X) = X^{n} + aX^{n−1} + b. Calculate Δ[1, α, α^{2}, …, α^{n−1}]. [1]

4. Let f(X) ∈ Q[ X] be an irreducible polynomial of degree n, and let K = Q(α) where f(α) = 0. Also let Δ = Δ_{K} denote the discriminant of K.

(i) Suppose n = 2 or n = 3. Show that Δ > 0 if and only if all the roots of f(X) = 0 are real. [1] (ii) Suppose n = 4. Show that Δ > 0 if and only if all the roots of f(X) = 0 are real, or none of the roots of f(X) = 0 are real. [2] (iii) In general, show how the sign of Δ depends on n and the number of real roots of f(X) = 0. [3]

5. Apply question 2 to give an alternative proof that O_{K} = Z[ ^{3} \sqrt{2} ] where K = Q( ^{3} \sqrt{2}). Also find O_{K} for K = Q( ^{3} \sqrt{3}), K = Q( ^{3} \sqrt{10}) and K = Q( ^{3} \sqrt{12}). [2,3,4]

6. For the following quadratic fields K, show that O_{K} is norm-Euclidean: Q(\sqrt{3}), Q(\sqrt{5}), Q(\sqrt{6}), Q(\sqrt{7}), Q(\sqrt{13}). [2,3] [Hint: first show that it suffices to show that if ξ = x + y\sqrt{Δ} with Δ = Δ_{K}, x, y ∈ Q, 0 ≤ x ≤ \frac{1}{2}, 0 ≤ y ≤ \frac{1}{4}, there exists α ∈ O_{K} with |N(ξ − α)| < 1. Then show that we can take α = 0, 1 or −1.]

7. Show that for the following quadratic fields K, O_{K} does not have unique factorization: Q(\sqrt{−5}), Q(\sqrt{−10}), Q(\sqrt{−15}), Q(\sqrt{10}), Q(\sqrt{15}). [2,3]

8. Find all units in O_{K} for K = Q(\sqrt{−1}), K = Q(\sqrt{−2}) and K = Q(\sqrt{−3}). [1] What can you say in general about the units of K = Q(\sqrt{−m}) for m > 0? [2]

9. Find all solutions of x^{2} + 4 = y^{3} with x and y integers. [3] 10. Let d be a negative squarefree number with d ≢ 1 (mod 4) and suppose that d has more than one prime factor. Show that I_{p} = ⟨p, \sqrt{d}⟩ is nonprincipal for each prime factor p of d, but that I_{p}^{2} is principal. [3]

11. Let I and J be ideals of a ring R. Show that if I + J = R, then IJ = I ∩ J. [2]

12. Show in detail that if I and J are ideals of a ring R then

{∑_{i=1}^{n} α_{i}β_{i} : α_{i} ∈ I, β_{i} ∈ J, n ∈ N}

is an ideal of R. [2]

13. Show that if I = ⟨α_{1}, …, α_{r}⟩ and J = ⟨β_{1}, …, β_{s}⟩ are ideals of some ring R, then

IJ = ⟨α_{1}β_{1}, α_{1}β_{2}, …, α_{1}β_{s}, α_{2}β_{1}, …, α_{r}β_{s}⟩ .  [2]

14. Let R = Z[ \sqrt{−6} ], and let I = ⟨2, \sqrt{−6}⟩. Show that 2 ∈ I^{2}, yet there are no α, β ∈ I with 2 = αβ. [2]

15. Let α = \frac{1}{2}(1 + \sqrt{−5}) and R = Z[ α] be the ring of integers of Q(\sqrt{−5}). Let I = ⟨2, α⟩, J_{1} = ⟨3, α⟩ and J_{2} = ⟨3, 1 − α⟩ be ideals of R. Show that I, J_{1} and J_{2} are all non-principal ideals, but that each of I^{2}, IJ_{1}, IJ_{2} and J_{1}J_{2} is a principal ideal. [3]

16. Let I and J be fractional ideals of O_{K}. Show that I + J, I ∩ J and IJ are also fractional ideals of O_{K}. [1] Show that (I : J) = {η ∈ K : ηJ ⊆ I} is a fractional ideal of O_{K}. [2] Assuming that J has an inverse, show that (I : J) = IJ^{−1}. [3]

17. Let J be a fractional ideal of O_{K}. Show that J is an ideal of O_{K} if and only if J ⊆ O_{K}. [1]

18. Let f = α_{0} + α_{1}X + ⋯ + α_{n}X^{n} be nonzero element of O_{K}[ X]. The ideal ⟨α_{0}, α_{1}, …, α_{d}⟩ is called the content c(f) of f. The polynomial f is primitive if c(f) = ⟨1⟩. Prove that if f and g are primitive then so is fg. [2] More generally prove that c(fg) = c(f)c(g) for any nonzero f and g. [4]

RJC 11/2/2000