Algebraic Number Theory: Problem sheet 3 Please hand in as many solutions as possible by the end of term and the remainder at the start of next term

The estimated difficulty (from 1 to 5) is given at the end of each problem 1. Let K = Q( ^{3} \sqrt{175}). Find Δ(1, ^{3} \sqrt{175}, ^{3} \sqrt{245}). [3]

Prove that 1, ^{3} \sqrt{175}, ^{3} \sqrt{245} forms an integral basis for O_{K}. [4] (Hint: use question 2 of sheet 2.)

Show that O_{K} has no integral basis of the form 1, δ, δ^{2}. [5]

2. Let m ≠ 1 be a squarefree integer congruent to 1 modulo 4, and set K = Q(\sqrt{m}). Let p be a prime number. Show that the prime ideal factorization of ⟨p⟩ in O_{K} is given as follows:

• if p | m then ⟨p⟩ = P^{2} for some prime ideal P;

• if p = 2 and m ≡ 1 (mod 8) then ⟨2⟩ = P Q with P and Q distinct prime ideals;

• if p = 2 and m ≡ 5 (mod 8) then ⟨2⟩ is prime;

• if p is odd, m ≢ 0 (mod p) and m is congruent to a square modulo p, then ⟨p⟩ = P Q with P and Q distinct prime ideals;

• if p is odd and m is not congruent to a square modulo p, then ⟨p⟩ is prime. [2]

3. Let K = Q(\sqrt{m}) be a quadratic field with discriminant Δ. Confirm that a prime p ramifies in K (that is ⟨p⟩ is the square of an ideal in O_{K}) if and only if p | Δ. [2]

4. Let K = Q(\sqrt{m}) be a quadratic field and define σ by σ(a + b\sqrt{m}) = a − b\sqrt{m} for a, b ∈ Q. Let I be an ideal of O_{K}. Show that σ(I) = \{σ(α) : α ∈ I\} is also an ideal of O_{K}. [1]

Suppose that I = ⟨α, β⟩. Show that Iσ(I) = ⟨n⟩ where n is the greatest common divisor of the integers ασ(α), βσ(β) and ασ(β) + βσ(α). Deduce that Iσ(I) = ⟨N(I)⟩. [3]

5. Retain the notation of the previous question. Let I be a nonzero ideal of O_{K}. Show that σ(I) = I if and only if I has the form I = ⟨n⟩ P_{1} ⋯ P_{k}, where n ∈ N, k ≥ 0, P_{j}^{2} = ⟨p_{j}⟩ where p_{j} is a prime number, and all the p_{j} are distinct. [3]

6. Retain the notation of the previous two questions. Let β ∈ K. Show that N(β) = 1 if and only if β = γ/σ(γ) for some nonzero γ ∈ K. [Hint: consider 1 + β.] Show further that we may assume that γ ∈ O_{K}. [2] 7. A result from number theory states that if p is an odd prime, then 2 is a square modulo p if and only if p ≡ 1 or 7 (mod 8) and −2 is a square modulo p if and only if p ≡ 1 or 3 (mod 8). Deduce that an odd prime p can be written in the form x^{2} + 2y^{2} with x and y integers if and only if p ≡ 1 or 3 (mod 8). State and prove the corresponding result for representations of primes as x^{2} − 2y^{2}. [3]

8. Let K = Q( ^{3} \sqrt{2}). Factorize ⟨2⟩, ⟨3⟩, ⟨5⟩, ⟨7⟩, ⟨11⟩, ⟨13⟩, ⟨17⟩ and ⟨19⟩ into prime ideals in O_{K} = Z[ ^{3} \sqrt{2} ]. [2]

9. Let I = P_{1}^{a_{1}}P_{2}^{a_{2}} ⋯ P_{r}^{a_{r}} and J = P_{1}^{b_{1}}P_{2}^{b_{2}} ⋯ P_{r}^{b_{r}} where the P_{j}s are distinct prime ideals of O_{K}. Show that I + J = P_{1}^{c_{1}}P_{2}^{c_{2}} ⋯ P_{r}^{c_{r}} and I ∩ J = P_{1}^{d_{1}}P_{2}^{d_{2}} ⋯ P_{r}^{d_{r}} where c_{j} = min(a_{j}, b_{j}) and d_{j} = max(a_{j}, b_{j}). [3]

10. Let P be a prime ideal of O_{K}. For each non-zero α ∈ O_{K} define v(α) as follows: let v(α) be the largest integer n with α ∈ P^{n}. Show that this definition makes sense, and that v has these properties: if α and β are nonzero elements of O_{K} then v(αβ) = v(α)+v(β) and v(α+β) ≥ min(v(α), v(β)) provided that α + β ≠ 0. [2]

11. The following result is proved in Stewart & Tall (chapter 5): if I is an ideal of O_{K} then I = ⟨α, β⟩ for some α, β ∈ I. Read the proof, and write out a version in your own words, incorporating the proofs of any preliminary results needed. [3]

12. Let p be a prime number, ζ = exp(2πi/p) and K = Q(ζ). Let q be a prime number with q ≠ p. Show that if O_{K} has a prime ideal of norm q, then ⟨q⟩ = Q_{1}Q_{2} ⋯ Q_{p−1} where the Q_{j} are distinct prime ideals, all of norm q. Also show that this occurs if and only if q ≡ 1 (mod p). [3]

RJC 29/2/2000