Final Exam¶

Symbol: In this papar, \(\mathrm{i}\) means \(\sqrt{-1}\), \(\omega\) means \(\dfrac{-1 + \sqrt{-3}}{2}\). When \(K = \mathbb{Q}(\sqrt{d})\), \(\mathcal{O}_K\) means the ring of integers of \(K\). And \(p\) is always an odd prime.

Attention: Promblem 1, 2 and 3 are the required problems. Problem 4, 5, 6 and 7 are the optional problems. You should choose one of 4, 5 and the other from 6, 7.

Question

(1) (i) Define \(\mathbb{Z}[\mathrm{i}]\).
(ii) State the Division Theorem in \(\mathbb{Z}[\mathrm{i}]\).
(iii) Give all the integer solutions of \(x^3 = y^2 + 1\).

(2) Does the equation

\[ x^2 \equiv 13 \pmod {43^2 \times 61^3} \]

have a solution? Clarify your answer.

(3) (i) Give the fundamental solution of the standard Pell equation \(x^2 - 2y^2 = 1\).
(ii) Give the complete solution of the standard Pell equation \(x^2 - 2y^2 = 1\).
(iii) Give the complete solution of the generalized Pell equation \(x^2 - 2y^2 = 7\).

(4) Prove that the group \((\mathbb{Z}/p^2\mathbb{Z})^{\times}\) always has a generator. (Hint: Consider the \(\bmod p\) map \(\varphi: (\mathbb{Z}/p^2\mathbb{Z})^{\times} \rightarrow (\mathbb{Z}/p\mathbb{Z})^{\times}\) which send \(n \bmod p^2\) to \(n \bmod p\).)

(5) \(K = \mathbb{Q}(\sqrt{-5})\).
(i) What is \(\mathcal{O}_K\)?
(ii) Give an example that the number in \(\mathcal{O}_K\) cannot be uniquely factorized into irreducible elements and clarify your answer.
(iii) Factorize the ideal \((6)\) into prime ideals in \(\mathcal{O}_K\) and compute the norm of each prime ideal.

(6) Give the complete integer solutions of equation \(x^2 + y^2 = 2z^2\).

(7) (i) Prove that if \(p \equiv 1 \pmod 3\), then \(p\) is not a prime in \(\mathbb{Z}[\omega]\).
(ii) Under which condition, \(x^2 + xy + y^2 = p\) has a solution in \(\mathbb{Z}\)?
(Hint: Consider the process we prove the equavialet condition of \(x^2 + y^2 = p\) has a solution in \(\mathbb{Z}\).)