Problem 1 (yau-contest 2022)
Section: Algebra and Number Theory

(a) Let $p(x) = a_n x^n + \cdots + a_1 x + a_0 \in R[x]$ be a polynomial over an integral domain $R$. Let $K$ denote the fraction field of $R$. Suppose $a/b \in K$ is a root of $p(x)$, where $a, b \in R$ and are relatively prime. Then, show that $a \mid a_0$ and $b \mid a_n$.

(b) Prove that $\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \mathbb{Q}(\sqrt{2} + \sqrt{3})$.

Problem 2 (yau-contest 2022)
Section: Algebra and Number Theory

Let $R$ be an integral domain with the fraction field $K$. An $R$-module $P$ is projective if there is an $R$-module $Q$ such that $P \oplus Q \cong F$ for some free $R$-module $F$. A fractional ideal $A$ is an $R$-submodule of $K$ such that $A = d^{-1}I$ for some ideal $I$ of $R$ and a nonzero element $d \in R$. A fractional ideal $A$ is called invertible if $AB = R$ for some fractional ideal $B$.

Show that an invertible fractional ideal $A$ is a projective $R$-module.

Problem 3 (yau-contest 2022)
Section: Algebra and Number Theory

Give a direct proof that the Lie algebra $\mathfrak{sl}(4, \mathbb{C})$ is isomorphic to the Lie algebra $\mathfrak{so}(6, \mathbb{C})$. (You should construct a Lie algebra homomorphism and prove that it is an isomorphism; you should not use Dynkin diagrams or the classification theory of simple Lie algebras.)

Problem 4 (yau-contest 2022)
Section: Algebra and Number Theory

Let $A = \mathcal{O}_K$ be the ring of integers of a number field $K$. Given a nonzero ideal $a \subset A$ and an arbitrary nonzero element $e \in a$, show that there exists $b \in a$ such that $a$ and $b$ generate $a$ (in particular, every ideal is 2-generated).

Problem 5 (yau-contest 2022)
Section: Algebra and Number Theory

Let $p$ be a prime number and $\zeta_p$ be a primitive $p$-th root of unity. Let $K = \mathbb{Q}(\zeta_p)$.

(a) Show that $\Phi_p(x) = \sum_{i=0}^{p-1} x^i$ is the minimal polynomial of $\zeta_p$ over $\mathbb{Q}$.

(b) Compute the trace $\mathrm{Tr}_{K/\mathbb{Q}}(1 - \zeta_p)$ and the norm $\mathrm{N}_{K/\mathbb{Q}}(1 - \zeta_p)$.

(c) Show that $(1 - \zeta_p)\,\mathcal{O}_K \cap \mathbb{Z} = p\mathbb{Z}$ and deduce that for all $y \in \mathcal{O}_K$, we have $\mathrm{Tr}_{K/\mathbb{Q}}(y(1 - \zeta_p)) \in p\mathbb{Z}$.

(d) Determine explicitly the ring of integers of $K$.

Problem 6 (yau-contest 2022)
Section: Algebra and Number Theory

Let $\theta \in \overline{\mathbb{Q}}$ be a root of the polynomial $f(X) = X^3 + 12X^2 + 8X + 1$. Let $K = \mathbb{Q}(\theta)$.

(a) Let $g(X) = X^3 + pX + q \in \mathbb{Z}[X]$. Compute the discriminant $\operatorname{disc}(g)$ of $g(X)$ in terms of $p, q$.

(b) Show that $f(X)$ is irreducible over $\mathbb{Q}$.

(c) Compute the discriminant $d_K(1, \theta, \theta^2)$. Please provide necessary details.

(d) For any arbitrary number field $F$ of degree $n$, let $a_1, a_2, \ldots, a_n \in \mathcal{O}_F$. Find and verify a sufficient condition in terms of the discriminant $d_F(a_1, \ldots, a_n)$ that the $a_1, \ldots, a_n$ form an integral basis of $F$.

(e) Write down an explicit integral basis of $K$ in terms of $\theta$ by using the above sufficient condition you have found. Please justify your arguments.