Tagged Questions
3
votes
0answers
59 views
Intuition behind “Non-Archimedean” — two senses of “non-archimedean”.
There appear to be two senses of the qualifier "Archimedean" for fields. One is for ordered fields, and one is for "valued fields" (fields with an absolute value function defined). In the first case, ...
2
votes
2answers
80 views
The composite of all unramified extensions inside an algebraic closure
I'm reading Ch.II, $\S$ 7 of Neukirch's Algebraic Number Theory and I'd be really grateful if someone could help me understand the following:
Let $K$ be a complete valued field wrt a non-archimedean ...
1
vote
0answers
130 views
Non-Archimedean Fields
Are there non-Archimedean fields without associated valuation or being a non-archimedan field implies it is a valuation field?
I understand that a non-Archimedean field is a field which does not ...
0
votes
1answer
139 views
What is a valuation associated to an ordering on a field?
If $(K,\leq)$ is a totally ordered field with $P\!=\!\{\alpha\!\in\!K;\, 0\!\leq\!\alpha\}$, how is the valuation associated to $P$ defined?
I was searching through Prestel & Delzell's Positive ...
0
votes
0answers
46 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 ...
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 ...
13
votes
1answer
306 views
Why is $\mathbb{C}_p$ isomorphic to $\mathbb{C}$?
I know that two closed fields of caracteristic $0$ and uncountable are isomorphic iff they have the same cardinality. But I don't know why $\mathbb{C}_p$ has the same cardinality as $\mathbb{C}$. Can ...
