For questions related to valuation functions on a field.
-1
votes
1answer
112 views
Discrete Valuation Rings problem 2
An order function on a field $K$ is a function $\phi:K\to \mathbb{Z} \cup {\{\infty}\}$ satisfying:
i) $\phi(a) = \infty$ if and only if $a=0$.
ii) $\phi(ab) = \phi(a) + \phi(b)$.
iii) ...
9
votes
4answers
265 views
Why does the equation $x^2-82y^2=\pm2$ have solutions in every $\mathbb{Z}_p$ but not in $\mathbb{Z}$?
I have been working on an exercise in H. P. F. Swinnerton-Dyer's book, A Brief Guide to Algebraic Number Theory. The question is like this:
Show that $x^2-82y^2=\pm2$ has solutions in every ...
5
votes
3answers
367 views
Algebraic Closure of Puiseux Series
Using the construction $R_N = K[t^\frac1N]$ $L_N = Quot(R_N)$ and $P = \bigcup_{N\in \mathbb{N}} L_N$ one automatically gets that the puiseux series are a field. Nevertheless they are also an ...
2
votes
2answers
35 views
Valuations, Isomorphism, Local ring
Let $x \in \mathbb{Q}, \, x \neq 0$, such that $x=p^r \frac{a}{b}, \quad a,b, r \in \mathbb{Z}, \quad p \nmid a, \quad p\nmid b$.
Let $v_p(x):=r$ and $v_p(0):= \infty$. Also,
$$\mathcal O_p= \left\{ ...
2
votes
1answer
152 views
Existence of an element of given orders at finitely many prime ideals of a Dedekind domain
Let $A$ be a Dedekind domain.
Let $P$ be a non-zero prime ideal of $A$.
Let $\alpha \in A$.
Let $k$ be a non-negative integer.
If $\alpha \in P^k$ and $\alpha\notin P^{k+1}$, we write $v_P(\alpha) = ...
6
votes
2answers
128 views
How many absolute values are there?
My question is the following: Are there algebraic norms on the fields $\mathbb{R}, \mathbf{Q_p}$ ''other'' than the absolute value, respectively $|\cdot|_p$?
Now phrasing more precisely: If generally ...
0
votes
0answers
45 views
What does it mean for a valuation to be normed?
I have a homework problem that uses the term: "a discrete normed non-trivial valuation" on a field.
We've defined the discrete trivial valuation in class, so that part is clear.
I think the natural ...
