# Questions tagged [separable-extension]

The tag has no usage guidance.

155 questions
Filter by
Sorted by
Tagged with
15 views

### Let $K=F(u)$ be a separable extension of $F$ with $u^m\in F$. Show that if the characteristic of $F$ is $p$ and $m=p^tr$, then $u^r\in F$.

Question: Let $K=F(u)$ be a separable extension of $F$ with $u^m\in F$ for some positive integer $m$. Show that if the characteristic of $F$ is $p$ and $m=p^tr$, then $u^r\in F$. (I think we also ...
48 views

### Is an algebraic extension of a separably closed field separably closed?

In the book "Rational points on varieties" by Bjorn Poonen, exercise 1.1 (a) states: "Prove that an algebraic extension of a separably closed field is separably closed." Is this ...
62 views

22 views

### Inseparable field extension of degree 2

I have searched for an example of a degree 2 field extension that is not separable. The example I see is the extension $L/K$ where $L=\mathbb{F}_2(\sqrt t), \ K=\mathbb{F}_2(t)$ where $t$ is not a ...
23 views

### Unseparable element becomes separable

Let $L/K$ be an extension of fields. Denote by $p$ the characteristic of $K$. One assumes that $p$ is a a prime. Let $a\in L$ not separable over $K$. Does it exist an integer $s$ such $a^{p^s}$ is ...
18 views

### Normal closure as the compositum of conjugates

An excerpt from Serge Lang's Algebra Chapter V $\S4$ p. 242. Let $E$ be a finite extension of $k$. The intersection of all normal extensions $K$ of $k$ (in an algebraic closure $E^\text{a}$) ...
44 views

111 views

### Solvability by radicals of a separable field of characteristic $p$

Preliminary Definitions A field $K$ is said to be solvable by proper radicals over $F$ if there exist fields $F_0,F_1,\dots,F_m$ such that $F=F_0\subseteq F_1\subseteq F_2\subseteq\dots\subseteq F_m$ ...
146 views

### If $L/K$ is a finite field extension and $L$ is perfect, then $K$ is perfect.

I'm aware that this question has been answered here. However, I wanna try and prove this directly: Let $L/K$ be a finite field extension, suppose $L$ is perfect and $\text{char}(K) = p > 0$. Show ...
156 views

### tower of separable extension

Let $K⊂L⊂M$ be a tower of fields. Let $L/K$ and $M/L$ be separable, is it true that $M/K$ is separable? I guess there are counterexamples, but I cannot point out them. Thank you for your kind help.[I ...
30 views

### Simple extension of purely inseparable extension

It is Albert "Modern Higher Algebra", chapter 7, section 9, exercises 5 and 6. Let $K$ be a field of degree $n$ over $F$ of characteristic $p$ such that every quantity $\alpha$ of $K$ is a ...
38 views

### Is $F$ over $\mathbb{F}_2(x)$ separable?

I want to determine whether the extension $F$ over $\mathbb{F}_2(x)$ is separable, where: F=\mathbb{F}_2(x)[t]/ \langle t^2+t+1 \rangle \quad \mbox{and} \quad \mathbb{F}_2 \mbox{ is the finite group ...
73 views

### If char K=0 , then every irreducible polynomial is separable [duplicate]

The following statement was left as exercise in my Field Theory class. Consider a field K and f $\in K[x]$ an irreducible polynomial. f is separable in some splitting field of f over K if every root ...
51 views

### Show that there exists an integer $n$ such that $a+nb$ is a primitive element of the field $K=F(a,b)$.

Let $F$ be a subfield of $\mathbb{C}$ and $a,b \in \mathbb{C}$ be algebraic elements over $F$. Show that there exists an integer $n$ such that $a+nb$ is a primitive element of the field $K=F(a,b)$. We ...