# Questions tagged [ramification]

Ramification in algebraic number theory means prime numbers factoring into some repeated prime ideal factors.

275 questions
Filter by
Sorted by
Tagged with
1 vote
91 views

### $\left(i\sqrt {\frac{21 \ + \ 5\sqrt{5}}{2}}\right)^n \notin \Bbb{Q}(ζ_{11})$ for all positive integer $n$

I want to prove $\left(i\sqrt {\frac{21 \ + \ 5\sqrt{5}}{2}}\right)^n$ does not lie in $\Bbb{Q}(ζ_{11})$ for all positive integer $n$. This problem arises from arithmetic geometry, but this problem ...
• 4,297
28 views

### If $Q \vartriangleleft \Bbb Z[i]$ lies over $(p) \vartriangleleft \Bbb Z$ where $p\in \Bbb Z\setminus\{2\}$, then $e(Q|p) = 1$.

Let $R = \Bbb Z$, $S = \Bbb Z[i]$, and $p \in R\setminus\{2\}$. If the ideal $Q \vartriangleleft S$ lies over $(p) \vartriangleleft R$, then $e(Q|p) = 1$. The primes (non-zero prime ideals) in $S$ ...
• 11.4k
1 vote
67 views

### Construct an extension of a number field of given degree where a given set of prime splits completely

I know that discriminant tells exactly which prime ramify in an extension, and it helps to construct extensions where a certain set of primes will ramify. But I don't know how to construct extensions ...
• 401
13 views

48 views

### For a finite extension $K/\mathbb{Q}_2$, the extension $K(\sqrt{-1})/K$ is always totally ramified

Let $K$ be a finite extension of the $2$-adic numbers $\mathbb{Q}_2$, and suppose that $-1$ is not square in $K$. Write $K(i)$ for the quadratic extension $K[X]/(X^2 + 1)$, where $i^2 = -1$. Is it ...
• 3,647
1 vote
86 views

• 23
42 views

### To find an element of galois group which sends one prime ideal above $p$ to another prime ideal above $p$

Let $L/K$ be finite galois number field extension and $p$ be prime of (ring of integers of) $K$. Let $G=Gal(L/K)$. It is well known that $G$ acts transitively on the set of all prime ideals of $L$ ...
• 4,297
1 vote
46 views

### $L/K$ is unramified extension implies corresponding local extension $L_P/K_p$ is unramified?

Let $K$ be a number field and $L/K$ be finite galois extension. Let $p$ be a prime ideal of ring of integers of $K$. Let $P$ be a prime ideal above $p$. Let's think about corresponding local extension ...
• 4,297
62 views

• 4,297
144 views

• 1,130
127 views

### Example of degree $n$ ramified, but not totally ramified extension

I'm looking for an example of degree $n$ ramified but not totally ramified example over $\Bbb Q_p$. I can find degree $n$ ramified extension, for example, $\Bbb Q_p({p^{1/n}})/\Bbb Q_p$. $p^{1/n}$'s ...
• 4,297
40 views

### Sufficient condition of $K'/K$ is ramified extension

Let $K$ be a local field and $K'$ be it's finite extension. And there exists $a∈K'$ such that $v(a)$ is not integer. Then, ramification index $e$ of $K'/K$ is at least 2, in other words, $K'/K$ is ...
• 4,297
1 vote
54 views

### Ramifies as well as split

Can somebody give an example of a finite Galois extension of $\mathbb{Q}$ where a rational prime $p$ ramifies ( some (equivalently every) prime lying over $p$ has ramification index $> 1$) as well ...
• 190
1 vote
I'm self-studying finite ramification theory, and I found the following problem: Let $\Phi$ be a semistandard Ehrhart form (i.e. its entries are weakly increasing). Prove that $\Phi$ has at least one ...