# Tagged Questions

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### $E/F$ is a finite field extension $\implies$ $\text{Aut}_F E$ is finite?

Question: If $E/F$ is a finite field extension, then is $G :=\text{Aut}_F E$ finite? Attempt: Let $\mathcal{B} := \{x_1, \ldots, x_n\}$ be a basis for $E/F$. Then $\sigma \in G$ is completely ...
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### Easy question about tower of fields

If $E/F/G$ is a tower of fields and $[E:G]<\infty$, then does $[F:G]<\infty$? I suspect the answer to be "yes", but somehow the fact that a basis of $E$ over $G$ might have vectors in ...
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### Denseness of algebraic and transcendental elements in R [NBHM 2014]

Which of the following statements are true? a. Algebraic numbers over Z are dense in R. b. Transcendental numbers over Z are dense in R. Let me write what I did. For a), the concerned set is a subset ...
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### What are the root of $x^3 - 2$ $\in \mathbb{R}[x]$? [closed]

From the given polynomial it is evident that we have to find a +ve number in $\mathbb{R}$ such that the cube is 2 if it exists. There is one and that is $\sqrt[3]{2}$. How to find the other roots in ...
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### Write Galois group as semidirect-product

Consider the polynomial $x^7-13 \in \Bbb{Q}(x)$. Find Galois group and write it as a semi-direct product. Edit: So here is what I have done. I found the dimension of the splitting field over ...
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### Cubic resolvent of quartic.

Where does the cubic resolvent of quartic come from? I would like to know its derivation since I am having a hard time memorising the formulas.
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### Field that contains $(\mathbb{Z}/p^n\mathbb{Z})^*$

Let $p$ be an odd prime. There exists a field $F$ that contains an isomorphic copy of $(\mathbb{Z}/p^n\mathbb{Z})^*$ as a multiplicative subgroup? Clearly if $n=1$ we can take $F=\mathbb{F}_p$, but ...
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### Finding degree of an extension

Find the degree of the field extension $\mathbb{Q}[\sqrt[3]{2},\sqrt[3]{3}]$ over $\mathbb{Q}$. My approach: Call the desired degree $n$. Clearly, $3|n$ and $n\leq 9$. So possible values of $n$ are ...
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### How do I show that both the additive and multiplicative groups of an infinite field are non-cyclic? [duplicate]

I've tried mimicking the proof in case of $\mathbb Q$ to deal with the characteristic zero case, but can't do the characteristic $p$ case. Can someone give a solution to that end?
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### Field Theory : What is wrong with this “homomorphism”?

Let $E/F$ be a field extension and $\alpha \in E$ be algebraic over $F$. Let $m(x) \in F[x]$ be irreducible and such that $m(\alpha) \neq 0$. Define $$\phi : F(\alpha) \to F[x]/(m)$$ by ...
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### Proof Unicity of Splitting Field

Context : Definition : We say $f(x) \in F[x]$ splits over the field extension $E/F$ if $$f(x) = c (x-\alpha_1)\cdots(x-\alpha_n)$$ for some $c, \alpha_1, \ldots, \alpha_n \in E$. A splitting ...
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### Infinite algebraic extension of $\mathbb{Q}$

I have this problem in a exercise list: "Prove that $K=\mathbb{Q}(2^{\frac{1}{2}},2^{\frac{1}{3}}, 2^{\frac{1}{4}}, \ldots)$ is an algebraic extension, but not a finite extension of $\mathbb{Q}$." ...
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### clarification of algebraic closure and algebraically closed field

Definition of Algebraic closure: An extension K of F is called an algebraic closure of F if a) F $\subset$ K is algebraic b) K is algebraically closed given the above definition I have been trying ...
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### $F(x,y)$ over $F$ is not simple

Let $F$ be a field. Let $x,y$ two algebraically independent indeterminates. Show that $F(x,y)/F$ is not a simple extension. Attempt: I tried by contradiction, assuming that $F(t)=F(x,y)$ and writing ...
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### When $\mathbb{Q}(\alpha+\beta)=\mathbb{Q}(\alpha, \beta)$?

Let $\alpha, \beta$ be algebraic numbers over $\mathbb{Q}$. Which necessary and sufficient conditions are known such that $$\mathbb{Q}(\alpha+\beta)=\mathbb{Q}(\alpha, \beta) \text{ ?} \tag{\ast}$$ ...
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### Why is ring of integers $\mathcal O_K$ called ring of integers - what properties of $\mathbb{Z}$ does it inherit?

I was wondering why ring of integers $\mathcal O_K$ for field $K$ is called ring of integers. Definition says that elements in this ring will be a solution for monic equation with coefficients ...
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### On the Characterization of $F[\alpha]$ for a Field Extension $E/F$.

Let $E/F$ be a field extension. Reading a proof that $$\alpha \text{ is algebraic over } F \implies [F[\alpha]:F] < \infty,$$ one might begin by considering an element $f(\alpha) \in F[\alpha]$ ...
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### Give an example of a field where -1=1

The question is to find a counterexample to the following: In every field $F$, $-1$ is not equal to $1$. My intuition leads me to integers modulo $1$. Is this correct, are the integers modulo $1$ a ...
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### Proof of a Field Extensions Theorem

Consider the following result. Theorem : Let $E/F$ be a finite field extension of degree $n$ and let $V$ be a vector space over $E$. Then $$\dim_F V = [E:F] \dim_E V.$$ Now, it seems like a ...
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### A field $K$ is an algebra

I learned this definition of an algebra recently. The definition is: A vectorspace $V$ over a field $K$ is is an algebra if there exists $K$-bilineair map $\varphi\colon V\times V\rightarrow V$ which ...
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### Field of prime characteristic over two indeterminates

Let $F$ have prime characteristic $p$ and let $E = F(Y,Z)$, where $Y, Z$ are indeterminates. Let $L=F(Y^{p} , Z^{p})$ $\subseteq E$. a. Show that $\alpha^{p} \in L$ for all $\alpha \in E$. b. Show ...
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### What would be a prime element in the field of rational numbers?

It is clear to understand what prime elements will be in case of ring of integers and many other rings. However, I find it confusing in case of fields, more specifically field of rational numbers. So ...
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