Questions tagged [extension-field]

Use this tag for questions about extension fields in abstract algebra. An extension field of a field K is just a field containing K as subfield, but interesting questions arise with them. Use this topic for dimension of extension fields, algebraic closure, algebraic/transcendental extensions, normal/separable extensions... This tag often goes along with the abstract-algebra and/or field-theory tags.

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Show that characteristic polynomial of $Ty = ay$ is power of minimal polynomial of $a$

Let $K/F$ be a finite, separable, algebraic field extension and let $T: K\to K, Ty = ay$. Show that $p = m^n$ where $p$ is $T$'s characteristic polynomial and $m$ is $a$'s minimal polynomial. $m$ ...
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number theory - extension field [closed]

Let $\gamma$ be an algebraic integer such that $\mathbb{Q} (\cos{2\pi/9}) \subset \mathbb{Q} (\gamma)$. can we say $\sin(2\pi/9)$ is in $\mathbb{Q} (\gamma)$?
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1 answer
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bijection between $Aut_F(E)$ and $Hom_F(E,\bar F)$

Question: Let $E/F$ be a normal extension,$\bar F$ be the algebraic closure of $F$.Denote the set of embeddings from $E$ to $\bar F$ fixing $F$ as $Hom_F(E,\bar F)$. The question is can we construct a ...
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Question about the minimal polynomial of an inseparable element in a field

Let $K/F$ be a field extension. The characteristic of F is prime $p$. Suppose $\beta \in K$ is an inseparable element of degree $p$, show the minimal polynomial of $\beta$ in $F$.
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Question about separable extension and $f(x)=x^{p^n}-a$

Let $f(x)$ be an irreducible polynomial in F which the leading coefficient is 1 and its degree is larger than 2. Show that if $f(x)$ has the same root in its splitting field, then the characteristic ...
1 vote
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Proving $[\mathbb{Q}(m(\alpha)):\mathbb{Q}]\leq[\mathbb{Q}(\alpha):\mathbb{Q}]$

Suppose $m$ is the minimal polynomial of a set $S$ of complex numbers such that for a complex number $\alpha\not\in S$, $\alpha$ is a root of the derivative of $m$. I want to prove that $[\mathbb{Q}(m(...
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2 votes
1 answer
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A doubt on generalizing a result on Extension fields

I had a doubt in generalizing a result on extension of fields involving composites. Upon searching, I came across this Extension Degree of Fields Composite. We have a result that, if $L$ and $M$ are ...
1 vote
1 answer
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Galois Theory - field extensions and irreducible polynomials

Let $n\in\mathbb{N}$. Find the extension degree and a basis of $\mathbb{Q}(\sqrt[n]{3})$ over $\mathbb{Q}$. Prove that $\sqrt[3]{3}\not\in\mathbb{Q}(\sqrt[4]{3})$. Prove that $x^3-3$ is irreducible ...
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1 answer
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Atiyah-MacDonald: Proposition 7.9. Proof

In the second half of the proof of Proposition 7.9. in Atiyah-MacDonald (Introduction to Commutative Algebra) we have a field $F = k[y_1,...,y_s]$ as a k-algebra with each $y_j = f_j/g_j$ with $f_j$ ...
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Tensor Product of an Extension Field and Quotient of a Polynomial Ring

Let $F$ be a field, and $E$ be an extension of it. Let $p$ be a polynomial over $F$. We need to show that $$E \otimes_F \frac{F[x]}{(p)} \cong \frac{E[x]}{(p)}.$$ Here is what I thought : Consider the ...
1 vote
1 answer
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Normality of field extensions via tensor product

Suppose $L/K$ is a finite extension of fields. Some of the properties of the extension can be characterised via properties of the tensor product $L \otimes_{K} L$. For example, $L/K$ is separable iff ...
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1 vote
1 answer
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Are there other topological field structures on the complex numbers which make the inclusion of the reals a topological embedding?

We consider the field of reals $\mathbb{R}$ with its standard topological field structure. Let $F = \mathbb{R}[t]/(t^2 + 1)$, as a field extension of $\mathbb{R}$. We know this extension is isomorphic ...
1 vote
1 answer
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Finite extensions of number fields as coverings

In chapter II on completions in Serge Lang's Algebraic Number Theory, he says that it is useful to think of finite extensions of a number field as coverings. I am confused about it. I thought maybe he ...
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Galois group of a polynomial over $\mathbb{Q}$

Let $t = m^2+m+7$ such that $m \in \mathbb{Z}$. Show that the Galois group of $f(x) \in \mathbb{Q}[x]$ denoted by $G(f(x),\mathbb{Q})$ is $A_3$ with $f(x) = x^3-tx-t$. Proof: Note that $f(x)$ is a ...
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$a$ has no $p-$th root in $K$, then show that $X^{p^n}-a$ is irreducible in $K[x]$

Suppose $char(K)=p$. Let $a\in K$. If $a$ has no $p-$th root in $K$, then show that $X^{p^n}-a$ is irreducible in $K[x]$ for all positive integers $n$. What I assumed is that $b$ is a $p^n$th root of $...
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If $\alpha$ is separable if and only if $K(\alpha)=K(\alpha^{p^n})$

Let $Char F=p$, If $\alpha$ be algebraic over $K$. Show that $\alpha$ is separable if and only if $K(\alpha)=K(\alpha^{p^n})$ for all $n \in \mathbb{N}$. I tried a lot for the first implication. But ...
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2 answers
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Find the degree of the extension $\mathbb{Q}(\sqrt{11},\sqrt[3]{11}):\mathbb{Q}(\sqrt{11})$

I am trying to find the degree of the following field extension: $\mathbb{Q}(\sqrt{11},\sqrt[3]{11}):\mathbb{Q}(\sqrt{11})$. I know it must be at most three since $\sqrt[3]{11}$ is a root of $x^3-11$. ...
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1 answer
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Why totally positive elements of a number field have a root

I heard that in a number field $K$ (let say $K = \mathbb{Q}[X]/P$), the totally positive element (that are sent into $\mathbb{R}^+$ for all morphisms of fields $K->\mathbb{C}$) all have a ...
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Splitting field of a polynomial and its galois extension

I am studying splitting fields and Galois extensions, and would like a sample proof of the following problem. Consider $P(x)=x^4−2x^2+9 ∈ Q[X]$, how do I show $K = Q(i +\sqrt2)$ is the splitting field ...
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Differential algebraicity and transcendence degree

The question that was left unanswered during a lecture was For a differential field $K$ and $x$ an element of a differential field extension of $K$, is it necessarily the case that for any element $x$...
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2 answers
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Order of the Galois Group is $|G(E/F)| = [E:F_G]$ where $F_G$ is the field fixed by $G$

I was reading Lisl Gaal' book on Galois Theory and, on page 84, the author uses the result in the title without proving it, leaving the proof to the reader (see below): I want to prove the following: ...
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Question about Weil Pairing & the MOV attack

When I read any description of the Weil Pairing, it's described as a map between the additive groups $G1$ x $G2$ of an Elliptic Curve to a different group. $G1$ is the r-torsion group of the EC curve ...
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1 vote
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Understanding the proof of uniqueness of splitting fields.

I am self-studying Galois theory and I am now reading splitting fields.I am having difficulty in understanding the proof of the fact that any two splitting fields of a polynomial $f(X)\in F[X]$ are $F$...
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3 answers
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Minimal polynomial of $\sqrt{1+\sqrt{2}}$ over $\mathbb{Q}$

I am trying to determine the minimal polynomial of $\sqrt{1+\sqrt{2}}$ over $\mathbb{Q}$ and explain why does it have degree $4$. I found that the minimal polynomial is $f(x)=x^4-2x^2-1$. It is monic, ...
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1 answer
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Invertible Elements of a field

basic question about field theory I know that if $p(x)$ is an irreducible polynomial in $\mathbb{Q}$, then $$\frac{\mathbb{Q}[x]}{\langle p(x) \rangle}$$ is a field. But if $p(x)$ is irreducible over $...
1 vote
1 answer
63 views

Is every polynomial of even degree reducible after some field extensions of degree 2?

Given an irreducible polynomial $p \in \mathbb{Q}[x]$ we're interested in how it factors after repeated simple field extension of degree $2$. So we generate a chain of fields $F_{n+1} / F_{n}$, where $...
1 vote
1 answer
28 views

Composition of two simple field extension with coprime degree is simple?

Let $E/F$ be a simple field extension of degree $m$ and $L/E$ be a simple field extension of degree $n$, where $\gcd(m,n)=1$. Is it necessary that $L/F$ is simple? In the setting of characterstic $0$ ...
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1 answer
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Proving the Existence of an Intermediate Field with Special Properties in an Inseparable Field Extension

I am reading Algebraic Number Fields by Janusz. In the middle of one of his proofs on page 23, he makes the following claim: Suppose $K$ is a field and $L$ a finite field extension that is not ...
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1 answer
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Doubt in Lang's proof of solvable extensions forming a distinguished class of extensions

The context for this question is the same as that is described in How to see that $M$ is Galois over $k$ in Lang's proof that solvable extensions are a distinguished class? (Prop. VI.7.1, *Algebra*...
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5 votes
1 answer
187 views

Irreducibility over $\mathbb{Q}[\sqrt[n]{p}]$

Let $p$ be a prime and $n=p+1$. Then prove or disprove whether $x^{2p}+nx+n$ is irreducible over $\mathbb{Q}[\sqrt[n]{p}]$. This question appeared in our quiz today. But I couldn't even prove that ...
3 votes
2 answers
67 views

Finding irreducible polynomial over the rationals

Question : Let $f(x)=x^3-3x+3$ on $\mathbb{Q}$, and $\alpha$ is a complex root of $f(x)$. For $\beta=1-\alpha+{\alpha}^2 $, find the minimal polynomial of $ \beta$ over $\mathbb{Q}$, $\ irr(\beta,\...
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3 votes
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Determine degree of a field extension with degree 8

I am trying to determine the degree of the field extension $\mathbb{Q}\subset\mathbb{Q}(\sqrt[4]{3},i)$. For this I use the following: $[\mathbb{Q}(\sqrt[4]{3},i):\mathbb{Q}]=[\mathbb{Q}(\sqrt[4]{3},i)...
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2 votes
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Galois Theory: $ a = i\sqrt{2}+\sqrt{3}$.

Let $ α = i\sqrt{2}+\sqrt{3}$ and let $K:=\mathbb{Q}(i\sqrt{2},\sqrt{3})$. I have to find the minimal polynomial, the degree of the extension $K:\mathbb{Q}$ and thus prove that it is simple. I started ...
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Completion of a scalar extension in the metric topology

Let $V$ be a complex vector space (not necessarily finite dimensional). Let $V[[t]]\cong V\hat{\otimes}_{\mathbb C}\mathbb C[[t]]$ be the completion of $V\otimes_{\mathbb C}\mathbb C[[t]]$ with ...
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Regarding Algebraic Dependence

Let $\Omega \supset F $ be a field extension and $A,B\subset \Omega$ be two sets. Def $1$: $A$ is algebraically independent over $F$ if every finite subset of $A$ is algebraically independent over $F$...
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Why the field extension $\mathbb{C}/\mathbb{Q}$ is not finitely generated? [closed]

I know the extension $\mathbb{C}/\mathbb{Q}$ is not finite, as $\mathbb{Q}$ is countable whereas $\mathbb{C}$ is not. However, how should I prove that the field extension $\mathbb{C}/\mathbb{Q}$ is ...
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Infinite algebraic field extension of a finite field is normal and separable

I am trying to prove that every infinite algebraic field extension $K$ of a finite field $\mathbb{F}$ is separable and normal. I know how to do it for finite $K$, but I am struggling to see why the ...
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Galois theory - proving two fields are equivalent

Define by $\mathbb{Q}(\alpha,\omega)$ to be the smallest subfield of $\mathbb{C}$ containing $\mathbb{Q}$ along with elements $\alpha,\omega$. Now let $\alpha=\sqrt[6]{5}$ and $\omega=\frac{1}{2}+\...
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1 answer
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Is there a field extension $E$ of $\mathbb R$ such that every formal power series with coefficients in $\mathbb R$ has a root in $E$?

Is there a field extension $E$ of $\mathbb R$ such that every formal power series with coefficients in $\mathbb R$ has a root in $E$? If so, can we describe its elements explicitly? The set of complex ...
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Absolute Galois groups and representations

I've thinking about the possibility how is the image of a two dimensional complex representations of absolute galois group of $\mathbb Q$, $G_\mathbb Q$. I thought I had a proof about what the image ...
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When is the product of two separable polynomials necessarily separable?

Obviously, this doesn't hold in general. I was watching this video and at 11:40 the creator claims that if $g = fh$ and $g$ is inseparable, while $f$ is separable, then $h$ must not be separable. This ...
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Why do two minimal polynomials of $\alpha$ over different extensions share the multiplicity of $\alpha$?

I am reading this PDF and concerned about Lemma 3. I will copy it here for completeness. Lemma 3. Assume that $K \subseteq E \subseteq L$ is a tower of finite field extensions. If an element $\alpha$ ...
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Possible Tetration extension for a specific interval (part 2)

So this is part 2 of a previous post, so I advise you to watch the first one to better understand this post. ("x tetration r" = ${^r}x$). Now, I'm gonna explain the method I use to extend ...
1 vote
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19 views

Are inseparable elements of an extension field closed under addition?

I know that both the separable and purely inseparable elements form fields. It seems very likely false that the inseparable elements form a field, but in a proof I'm developing the following fact ...
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Valuation rings over a field that is algebraic over a finite subfield, are there any?

I am studying an article by Koji Sekiguchi and am stuck at a corollary that states: $Zar(K) =\{K\} $ if and only if $K$ is an algebraic extension over a finite field. Where $Zar(K)$ denotes the set of ...
2 votes
1 answer
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Possible Tetration extension for a specific interval (part 1)

My friend and I have been developing an extension of tetration for non integer values. We managed to get definitions of extensions for : ${^r}x$. $x$>0. $r$ not equal to any whole number below -1. ...
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Why is an element separable only if it is of the form $\alpha^p$ for some $\alpha$?

Here is a problem from Allan Clark's Abstract Algebra book: Let $F$ be a field, $p > 0$ the characteristic of $F$ and $E$ a finite algebraic extension of $F$. Denote by $E^{(p)}$ the minimal ...
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3 votes
1 answer
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Degree of the minimal polynomial of $\sum\sqrt{2\pm\sqrt{2\pm\sqrt{2}}}$.

The polynomial $$f(x)= x^8-8x^6+20x^4-16x^2+2=((x^2-2)^2-2)^2-2=0$$ has $$y=\sqrt{2+\sqrt{2+\sqrt{2}}}$$ one of its roots. How do I determine the degree of its spliting field and how do I determine ...
1 vote
2 answers
102 views

Prove that there exists an infinite field $F$ such that $F$ is algebraic over a finite field, and $F$ is not algebraically closed [duplicate]

Prove that there exists a field $F$ such that $F$ is infinite $F$ is algebraic over a finite field $F$ is not algebraically closed. I can see that if condition (3) is removed, then the algebraic ...
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0 votes
1 answer
36 views

The splitting field on $\mathbb{Q}$ of $x^4+10x^2+5$ is $\mathbb{Q}[i\sqrt{5-2\sqrt{5}}]$

$u = i\sqrt{5-2\sqrt{5}}$, if $f$ is the minimal polynomial of $u$ on $\mathbb{Q}$ and $\mathbb{E}$ is the splitting field of $f$ on the rationals, then $\mathbb{E} = \mathbb{Q}[u]$. Now $f=x^4+10x^2+...

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