All Questions
Tagged with p-adic-numbers fields
10 questions
5
votes
1
answer
485
views
General algebraic result obtained from consideration on $\mathbb{Q}_p$
There are results in field theory which are obtained from, let's say, the complex numbers and then generalized to all algebraically closed fields.
For instance, the fact that a polynomial $P$ admits a ...
- 241
7
votes
0
answers
394
views
Do algebraic completion/topological completion of fields always terminate? If so, are they unique?
Take the field $\mathbb{Q}$, If we complete it topologically with respect to the Euclidean norm, we get $\mathbb{R}$, then if we complete it algebraically, we get $\mathbb{C}$.
On the other hand, the ...
- 171
3
votes
0
answers
118
views
Composition in function fields
Let $k=\mathbb F_q\left(\!\left(\frac1T\right)\!\right)$. One has the map: $\circ:k\times\{v\in k\mid\deg(v)>0\}\to k$ defined by $f\circ g=\sum_{n\ge-m}a_ng^{-n}$ where $f=\sum_{n\ge-m}a_n\frac1{T^...
- 3,998
7
votes
0
answers
154
views
On the isomorphism of the lattices of submodules of certain free modules
Let $K,L$ be two finite extensions of the $p$-adic field
$\mathbb{Q}_p$ of the same degree. Let $\mathcal{O}_K$ and
$\mathcal{O}_L$ be the ring of integers of these two fields, and let
$\mathcal{O}_K^...
- 50.8k
3
votes
0
answers
76
views
Continuous extension of the derivation in positive characteristic
Let $\Omega$ be the completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the topology induced by the valuation $-\deg$. Does there exist a derivation on $\Omega$ ...
- 3,998
2
votes
0
answers
140
views
Completing vs Extending a field
Given a field and a metric on it, consider the goal of completing it and extending it in order to get an algebraicly closed and complete field.
How should one proceed? Should one first complete it ...
- 1,638
5
votes
1
answer
898
views
p-adic expansion for elements in algebraic closure of p-adic numbers
In the following I will describe a proposal for the p-adic expansion of the elements of the algebraic closure $\overline{\mathbb{Q}_p}$ of $\mathbb{Q}_p$. My question is if this "conjecture" has been ...
- 596
0
votes
0
answers
231
views
Unexpected isomorphisms between "unrelated fields"
I read in the post Why worry about the Axiom of Choice ? that the existence of isomorphisms between $\overline{\mathbb{Q}_p}$, $p$ any prime, and $\mathbb{C}$, makes some worry about the Axiom of ...
- 4,605
6
votes
0
answers
514
views
Finite extensions of $\mathbb Q_p$
Is there any finite extension of $\mathbb Q_p$ which is not the completion of a finite extension of $\mathbb Q$ at some place over $p$?
Analogously in equicharacteristic, if $k=\overline {\mathbb F_p}$...
- 277
11
votes
1
answer
3k
views
Ring of Witt vectors and p-adics
This is probably an easy question, but I'm not able to figure it out.
Are the following the same:
Field of fractions of the ring of Witt vectors over the algebraic closure of $\mathbb{F}_p$
...
- 7,320