Articles of ramification

Number Theory: Ramification

The question is as follows. Let $K = \mathbb{Q}(\sqrt[m]{a},\sqrt[n]{b}) $, where $m,n,a,b$ are positive integers such that they are pairwise coprime. Assume that $[K:\mathbb{Q}]=mn$/ Prove that no prime numbers can totally ramify in $K/\mathbb{Q}$. I assume we would need to find such prime numbers that are ramified in $\mathbb{Q}(\sqrt[m]{a})/\mathbb{Q}$, $\mathbb{Q}(\sqrt[n]{b})/\mathbb{Q}$ respectively. I know if $p|\text{disc}(L)$, […]

Relationship between wild ramification and restriction in Galois extension

For each $n \geq 1$, we denote by $\mathbb{Q}_n$ the unique subfield of $\mathbb{Q}(\zeta_{p^{n+1}})$ for which $[\mathbb{Q}_n:\mathbb{Q}]=p^n$. Let now $K$ be a finite Galois extension of $\mathbb{Q}$ with the following properties: K contains $\mathbb{Q}_m$ for some $m \geq 1$; There exists an element $g \in Gal(K/\mathbb{Q})$ with the property that the order of $g$ is […]

wildly ramified extensions and $p$-power roots of unity

Let $K$ be a finite extension of $\mathbf{Q}_p$. I have this vague intuition that $K$ having a lot of wild ramification is closely related to $K$ being close to containing high degree $p$-power roots of unity. I was wondering if this intuition is true and if maybe the following result (which would formalise it) is […]

Intuition in studying splitting and ramification of prime ideals

I am trying to learn Algebraic Number Theory alone and I’m having serious trouble understanding the ramification and splitting of primes ideals in Galois extensions of a number field $L/K$. Some of my friends tell me lots of facts about these and define several structures and invariants, but I feel like the logical sequence of […]

$\mathbb{Q}(\sqrt{17})$ has class number $1$

Let $\alpha:=\mathbb{Q}(\sqrt[3]{17})$ and $K:=\mathbb{Q}(\alpha)$. We know that $$\mathcal{O}_K=\left\{\frac{a+b\alpha+c\alpha^2}{3}:a\equiv c\equiv -b\pmod{3}\right\}.$$ I have to show that $K$ has class number $1$, i.e. $\mathcal{O}_K$ is a PID. The Minkowski bound $\lambda <9$, so we should consider the primes $2, 3, 5, 7$. It’s easy to show that $2\mathcal{O}_K=\mathfrak{p}_1\mathfrak{p}_2$, with $\mathfrak{p}_1=(2, \alpha+1)$ and $\mathfrak{p}_2=(2, \alpha^2+\alpha+1)$ $3\mathcal{O}_K=\mathfrak{p}_3^2\mathfrak{p}_4$ (I can’t […]

Computing a uniformizer in a totally ramified extension of $\mathbb{Q}_p$.

Do you know how to compute a uniformizer of $\mathbb{Q}_p(\zeta_{p^n},p^\frac{1}{p})$? Where $\zeta_{p^n}$ is a primitive $p^n$-th root of 1 and $p$ is an odd prime.