试卷: 代数数论

12024-25 春

1.	Judgement (Just write true and false) (1) For any Dirichlet character $\chi$, the L-function $L_{\chi }(s)$ is holomorphic in the half plane $\mathrm{Re}s>0$; (2) $\frac{\sqrt{3}+\sqrt{5}}{2}$ is an algebraic integer; (3) For any number field $L$, $x\in {\mathcal{O}}_L\iff N_{L/\mathbb{Q}}\in \mathbb{Z}$; (4) The rational prime $p$ such that $x^2+1\equiv 0(\mathrm{mod}p)$ has solution has Dirichlet density $\frac{1}{2}$ in all the rational primes; (5) For any valuation field $(K,v)$ and a finite algebraic extension $L/K$, $v$ has a unique extension in $L$.
2.	Denote $f(x)=x^3-x^2-2x-8$, $K=\mathbb{Q}[x]/f(x)$, $\alpha$ the image of $x$ in $K$. (1) Prove: $f$ is irreducible over $\mathbb{Q}$; (2) How many primes of ${\mathcal{O}}_K$ are over prime $2$? (3) Compute $\mathrm{Disc}(1,\alpha ,{\alpha }^2)$; (4) Prove that $\beta \triangleq \frac{{\alpha }^2-\alpha }{2}$ is integral over $\mathbb{Z}$; (5) Prove: $\{1,\alpha ,\beta \}$ form an integral basis of $K/\mathbb{Q}$.
3.	Let ${\zeta }_7$ be the $7$-th root of unity, $K$ be the unique intermediate of $\mathbb{Q}({\zeta }_7)/\mathbb{Q}$ such that $[K:\mathbb{Q}]=3$. (1) Does there exists an embedding $\sigma :K\hookrightarrow \mathbb{C}$ such that the image of $\sigma$ is not in $\mathbb{R}$? (2) Classify by module $7$ which rational primes are ramified, inert and split in ${\mathcal{O}}_K$.
4.	(1) Prove: The series$$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{5}+\frac{1}{7}-\frac{1}{8}+\cdots$$converges conditionally in $\mathbb{R}$, and find its value; (2) In the $p$-adic number field ${\mathbb{Q}}_p$, prove that the series$$\sum _{n=1}^{\infty }n\cdot n!,\sum _{n=1}^{\infty }{n}^2\cdot (n+1)!$$converge and find their values.
5.	(1) Prove: Let $p$ be an odd prime, then ${\mathbb{Q}}_p$ has no $p$-th root of unity other than $1$; (2) Prove: ${\mathbb{Q}}_2$ has no $4$-th root of unity other than $\pm 1$. (3) Find all $m\in \mathbb{N}$ such that ${\mathbb{Q}}_p$ has all $m$-th root of unity. (4) Prove: ${\mathbb{Q}}_p$ is not isomorphic to ${\mathbb{Q}}_q$ for $p\ne q$.
6.	Let ${\mathbb{F}}_p(X)$ be the field of Laurent series. (1) Does ${\mathbb{F}}_p(X)$ has an archimedean valuation? (2) Classify all valutations on ${\mathbb{F}}_p(X)$ under equivalence.