All Questions
Tagged with ac.commutative-algebra field-extensions
14 questions
50
votes
0
answers
2k
views
How many algebraic closures can a field have?
Assuming the axiom of choice given a field $F$, there is an algebraic extension $\overline F$ of $F$ which is algebraically closed. Moreover, if $K$ is a different algebraic extension of $F$ which is ...
- 39.7k
37
votes
4
answers
12k
views
Finite extension of fields with no primitive element
What is an example of a finite field extension which is not generated by a single element?
Background: A finite field extension E of F is generated by a primitive element if and only if there are a ...
10
votes
1
answer
243
views
If $E_\text{sep}/F$ is normal, then must $E/F$ be normal?
This question has been asked in Math.StackExchange (see here) for more than a week and I even put a bounty on it. But still it hasn't been correctly answered (the current answer there was written by ...
- 452
9
votes
1
answer
313
views
Concerning $k \subset L \subset k(x,y)$
The following is a known result in algebraic geometry:
Let $k$ be an algebraically closed field of characteristic zero (for example, $k=\mathbb{C}$).
Let $L$ be a field such that $k \subset L \subset ...
- 2,837
6
votes
1
answer
336
views
Is the minimal polynomial of an algebraic formal Laurent series always separable?
Let $f(x)\in K((x))$ be an algebraic formal Laurent series and let $P(x,y)\in K(x)[y]$ be its minimal polynomial. Is $P(x,y)$
always separable? An example of non separable polynomial comes
from ...
- 385
3
votes
0
answers
111
views
A bound for $[\mathbb{C}(x,y,z):\mathbb{C}(p,q,r)]$, where $\operatorname{Jac}(p,q,r) \in \mathbb{C}^{\times}$
Y. Zhang (in his PhD thesis) and P. I. Katsylo proved the following nice result; the two proofs are different, see: Zhang's thesis and Katsylo's paper:
Let $f: (x,y) \mapsto (p,q)$ be a $k$-algebra ...
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2
votes
1
answer
202
views
Artin-Schreier theorem for rings (a little different)
Motivation:
Let me recall the well-known Artin-Schreier theorem (AST) for fields in a non-formal way; if $L$ is an algebraically closed field, and $K \subset L$ a subfield not 'much smaller' than $L$, ...
- 748
1
vote
1
answer
92
views
On analytic transcendence degree and Krull dimension for homomorphic images of power series rings
Let $k$ be a field of characteristic zero and $I$ be a radical ideal of $k[[x_1,\ldots, x_n]]$. Let $P$ be a minimal prime ideal of the reduced ring $R:=k[[x_1,\ldots, x_n]]/I$. Then, $R_P$ is a field ...
- 480
1
vote
1
answer
149
views
$F=\mathbb{C}(u,v)$ satisfying: For every $a,b \in \mathbb{C}[y],c,d \in \mathbb{C}[x]$: $\mathbb{C}(x,y)=F(ax+b)=F(cy+d)$
Let $u,v \in \mathbb{C}[x,y]$, where $u$ and $v$ are algebraically independent over $\mathbb{C}$ and $F=\mathbb{C}(u,v)$. Of course, $d:=[\mathbb{C}(x,y):F] < \infty$.
Denote the following ...
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1
vote
1
answer
88
views
Transcendence degree and Krull dimension for homomorphic images of power series rings
Let $k$ be a field of characteristic zero and $I$ be a radical ideal of $k[[x_1,\ldots, x_n]]$. Let $P$ be a minimal prime ideal of the reduced ring $R:=k[[x_1,\ldots, x_n]]/I$. Then, $R_P$ is a field ...
- 480
1
vote
1
answer
198
views
Shrinking the base field of an affine variety
This is a question on algebraic geometry/commutative algebra.
Let $K,L$ be fields of characteristics zero and let $K\subset L$ be a field extension (I am interested in the case when this is ...
- 21
1
vote
0
answers
59
views
If $E \subseteq F=k(x_1,\ldots,x_r)$, satisfies $E(x_1^{i_1},\ldots,x_r^{i_r})=F$, for every $(i_1,\ldots,i_r) \neq (0,\ldots,0)$, then $[F:E] \leq 2$
For $r \geq 2$, let $A_r=\mathbb{C}[x_1,\ldots,x_r]$,
$F_r=\mathbb{C}(x_1,\ldots,x_r)$ the field of fractions of $A_r$, and $E_r \subseteq F_r$ an arbitrary subfield of $F_r$ with $[F_r:E_r] < \...
- 2,837
1
vote
0
answers
360
views
A composition of a simple extension and a separable extension is simple
Let $K/L/M$ be a tower of finite field extensions with $K/L$ separable and $L/M$ simple (in the sense of being generated by a single element). How does one show that $K/M$ is also simple?
I know that ...
0
votes
1
answer
473
views
A subfield $R \subseteq \mathbb{C}(x,y)$ with 'many' generators $w$, $R(w)=\mathbb{C}(x,y)$
Let $R \subseteq \mathbb{C}(x,y)$ and assume that $R=\mathbb{C}(u,v)$, where $u,v \in \mathbb{C}[x,y]$ are algebraically independent over $\mathbb{C}$.
Here $\mathbb{N}$ includes $0$.
Assume that $R$ ...
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