# Tagged Questions

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### Is a finitely generated extension of a real closed field also real closed?

Let $\widetilde{\mathbb{Q}}$ be the field of real algebraic numbers, and consider $\widetilde{\mathbb{Q}}(\pi)$. My question: is $\widetilde{\mathbb{Q}}(\pi)$ a real closed field? Bonus karma points ...
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### A result on extension fields in linear algebra.

Let $F$ be a subfield of $E$, $A$ an element of $\mathcal{M}_F(m,n)$ and $b$ a vector in $F^m \subset E^m$. What is the easiest way to prove the following statement: if $Ax = b$ has a solution in ...
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### Galois group - extend homomorphism to automorphism

Let $K \subset L$ be a finite Galois extension, $M$ a field with $K \subset M \subset L$ and $G := \text{Aut}(L/K)$. I want to show that if $\sigma \, \colon M \longrightarrow L$ is a ...
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### There are finite distinct restrictions to a subfield

Consider the field extension $L\subseteq K\subseteq \mathbb C$ where $K/L$ is finite. I must show that the set $\{\sigma_{|K}\,:\,\sigma\in\operatorname{Gal}( \mathbb C/L)\}$ is finite, but I have ...
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### simple extension with algebraic over the field

Assume that $F$ is infinite, that $v,w \in K$ are algebraic over $F$, and that $w$ is a root of a separable polynomial in $F[x]$. How will I be able to prove that $F(v,w)$ is a simple extension of ...
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### Prove that if $a^2 + ab + b^2 = 0$ then $a = b = 0$?

We are given that $a, b \in F_{2^n}$ where $n$ is an odd +ve integer. Suppose $a^2 + ab + b^2 = 0$ then we have either $a = 2^n-b^2$ or $a+b = 2^n - b^2$. Which implies that $\sqrt{2^n -a} = +-b$ or ...
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### Prove that every extension of a finite field is normal

In book by Roman 'Field Theory' it is written that it is straightforward that every extension of a finite field is normal. However I just cannot see it. Can you help me with this problem? Thank
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### finite field extension and normality

If $K$ is an extension field of $F$ such that $[K:F]=2$. Then $K$ is normal? I know that if $[K:F]=2$ then $K=F(u)$ where $u$ is the root of $f \in F[x]$. But how do you prove that dimension $2$ ...
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### Rupture field of $X^p+T$ equals its splitting field [closed]

Let $K$ be a field of prime characteristic $p$. Let $P(X)=X^p+T$ be a polynomial from $K(T)[X]$. $P$ is irreducible over $K(T)$ by Eisenstein criterion. Show that a rupture field of $P$ is also a ...
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### Embedding a finite extension of $F(X)$ into a pure transcendental extension

If $F$ is any algebraically closed field, and $L \supset F(X)$ is a finite extension of the purely transcendental extension of $F$ of transcendence degree $1$, then can $L$ necessarily be embedded ...
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### Basis for field extension by an algebraic element

Is was wondering if, given a field F with a known basis and an element b which is algebraic over that field, it is possible to construct explicitly a basis for F[b], the extension of F by b. Suppose, ...
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### Why isn't $e^{\frac{2\pi i}{9}}$ an element of $\mathbb{Q}(\sqrt[9]{2}, e^{\frac{2\pi i}{3}})$?

The question is in the headline above. I need to know this, because I want to show that $\mathbb{Q}(\sqrt[9]{2}, e^{\frac{2\pi i}{3}})/\mathbb{Q}$ is not a normal extension and it's quite obvious I ...
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### Calculate the degree of a field extension

Find degree $[\mathbb{Q}(i,\sqrt{2}) : \mathbb{Q}]$ let $a = \sqrt{2}$ $a^2 = 2$ $\therefore a$ is a root of $q(x) = x^2 -2$, where $q(x)\in\mathbb{Q}(i\sqrt{2})[x]$ means degree of $a$ over ...
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### Show that two field extensions are the same

Can you help me with showing that these two field extensions are the same: $\mathbb{Q}(\sqrt{3}, \sqrt[3]{5})$ $\mathbb{Q}(\sqrt{3} + b\sqrt[3]{5})$, where $b\neq0$ is any rational number. Thanks ...
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### Find a subfield of $\mathbb{C}$ isomorphic to other field

Do you know, how I can find a subﬁeld of $\mathbb{C}$ isomorphic to $F = \mathbb{Q}(\sqrt[3]{7})$ such that $F \nsubseteq \mathbb{R}$? I don't even have clue, how I should start. Thanks
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### The Galois Group of $x^4 - 5x^2 + 6$

As the title suggests, I'm asked to describe the Galois group of the polynomial $x^4 - 5x^2 + 6 \in \mathbb{Q}[x]$ over $\mathbb{Q}$. I am pretty certain I have 95% of the problem completed. I'm just ...
I have a basic question about algebraic field extensions: How can I convert a multiple extension like $\mathbb{Q}(\sqrt{2},\sqrt{3})$ to a single (elementary) field extension (like ...