Questions tagged [separable-extension]
244 questions
12
votes
1 answer
The definition of the separable closure of a field
Could someone tell me the definition of the separable closure of a field $K$?
Furthermore, I would like to know whether it is a Galois extension of $K$. Also, why is this construction useful?
Mark Rodriguez
- 969
11
votes
1 answer
How to calculate separable closures of an algebraic extension?
In my study of fields, the notion of the separability of an algebraic field extension is one of the more slippery concepts I have encountered thusfar. What is particularly vexing to me is the notion of a separable closure. Let $E/F$ be an algebraic…
Holdsworth88
- 8,948
8
votes
2 answers
Ways of finding primitive element of separable extension $\Bbb{Q}(\sqrt[4]{2},i)$ over $\Bbb{Q}$.
Consider the field extension $L=\mathbb Q (\sqrt[4] 2 ,i)$ over $\mathbb Q$. This extension is separable as we know over a field of characterstic $0$. Now according to the primitive element theorem there exist an element $\gamma$ such that…
Theorem
- 8,359
8
votes
0 answers
What is an example of a non-simple finite extension $K/F$ such that the purely inseparable closure of $F$ in $K$ is simple?
The standard example of a finite extension that is not simple is to take $k$ to be a field of characteristic $p > 0$ and consider $k(x,y)$ over $k(x^p,y^p)$. In this case, the extension is purely inseparable too.
I was wondering the following:
Is…
user279515
7
votes
1 answer
Separable extension and tensor product
Let $F$ be a field and let $\bar{F}$ be an algebraic closure of $F$. $K/F$ is a finite extension of degree $d$. Could you please tell me why the following conditions are equivalent?
K/F is separable;
$K\otimes_F \bar{F}\simeq (\bar{F})^d$;
The ring…
Xiangzhen Zhang
- 1,156
7
votes
1 answer
Why every extension of a characteristic zero field is separable?
My problem is to prove that the splitting field of $f(X)=X^5+X+1$ is, in fact, a separable extension of a field $F$ with characteristic zero.
After some research, I found that every extension of a field with characteristic zero is separable.
The…
Mateus Rocha
- 2,746
7
votes
2 answers
How to compute a primitive element for the splitting field of $x^3-2 \in \Bbb{Q}[x]$?
Let $\alpha:=\sqrt[3]{2}\in\mathbb{R}$ and $\omega:=e^{2\pi i/3}\in\mathbb{C}$. Then the splitting field for the polynomial $x^3-2\in\mathbb{Q}[x]$ is $$\mathbb{Q}(\alpha,\omega\alpha,\omega^2\alpha)=\mathbb{Q}(\alpha,\omega).$$
Since $\mathbb{Q}$…
Drew Armstrong
- 1,023
6
votes
3 answers
Can a field extension still have "non-separability" above its maximal purely inseparable subextension?
Question 1
Let $E/F$ be an algebraic field extension. Let $K$ be the set of all elements of $E$ that are purely inseparable over $F$. Then, $E/K/F$ is a tower of fields, and $K/F$ is purely inseparable.
In this case, is $E/K$ always separable? If…
Rubertos
- 12,941
5
votes
1 answer
Proof of separability of polynomials without derivatives
Is there a known proof without differentiating that proves that all irreducible polynomials over $\mathbb{Q}$ are separable? (Or even better, for all fields of characteristic $0$.)
EDIT: As people seem to question this thread; I do know a proof with…
Qi Zhu
- 9,413
5
votes
2 answers
If $L/K$ normal and $H = \operatorname{Aut}(L/K)$, then $L/L^H$ is separable and $L^H/K$ is purely inseparable.
I need to prove the following:
Let $L/K$ be a normal field extension. Denote by $H=\operatorname{Aut}(L/K)$ the Galois group of the extension, and by $L^H$ the fixed field of $H$ in $L$. Prove that $L/L^H$ is separable, and that $L^H/K$ is purely…
Or Kedar
- 933
5
votes
3 answers
Show that if $K \subset L$, then the separable closure of $K$ in $L$ is a field
Let $K \subset L$ be a field extension. Consider the separable closure $K_s$ of $K$ in $L$ defined as
$$
K_s = \left\{ {x \in L \mid x \text{ is algebraic and separable over } K} \right\}
$$
Prove that $K_s$ is a field.
I know how to prove…
Joseph
- 233
5
votes
2 answers
$F/K$ finite extension, then $\exists !$ intermediate field $K \subset L \subset F$ such that $L/K$ separable and $F/L$ purely inseparable
I've completed an introductory course in Galois Theory, but feel my understanding of separability is poor. I think my confusions boil down to the following question:
What is the relationship between a separable extension $L/K$ and the space…
Matt
- 1,033
5
votes
2 answers
If $F^p = F$ and $E/F$ is algebraic, then $E/F$ is separable and $E^p = E$ : Corollary V.6.12 from Lang's *Algebra*
This is from Lang's Algebra (page 251)
Proposition 6.11 Let $E/F$ be a normal field extension. Let $E^G$ be the fixed field of $\operatorname{Aut}(E/F)$. Then, $E^G$ is purely inseparable over $F$ and $E$ is separable over $E^G$.
And below is a…
Rubertos
- 12,941
4
votes
1 answer
Function field over a perfect field can be generated by two elements
I have two questions about the following theorem:
Theorem: Let $K$ be a perfect field, $F$ a function field in one variable over $K$ (i.e., a finite algebraic extension of $K(t)$). Then there is $x \in F$ such that $F$ is finite and separable over…
Marktmeister
- 1,896
4
votes
0 answers
Why does separable degree behave unexpectedly in infinite extensions?
First of all, my definition of separable degree of $K/F$ is $[K:F]_s=|\operatorname{Hom}_{F-alg}(K,\bar F)|$ where bar denotes algebraic closure.
It is well-known that in finite field extensions $[K:F]_s|[K:F]$. However, it is also known that in the…
Tesla Daybreak
- 1,663