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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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21 views

Splitting field over K of an infinite set of polynomial

Suppose $F$ is a finite splitting field over $K$ of $X=\lbrace f_i(x)\rbrace_{i\in I}$, some infinite set. Is there necessarily a finite set $Y\subseteq X$ such that $F$ is a finite splitting field of ...
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1answer
24 views

$K$ is intermideate field between $F$ and $F(x)$. then $\operatorname{dim}F(x)$ over $K$ is finite.

Let $F$ be a field and let $F(X)$ be the field of rational functions with coefficients in $F$. Let $K$ be any field such that $F\subseteq K\subseteq F(X)$ and $K\neq F$. Prove that $[F(X):K]\lt\infty$....
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2answers
45 views

What is the index of field extension $\mathbb{C}/\mathbb{R}$?

What is the index of field extension $\mathbb{C}/\mathbb{R}$? I know that the answer is $2$, but if so, that means $\mathbb{C}/\mathbb{R}=\{\mathbb{R}, i+\mathbb{R}\}$, and how come $5i+\mathbb{R}=i+\...
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1answer
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Normal extension and effect of field automorphisms

Definition 1. An extension $E$ of a field $k$ is called normal extension if (i) $E$ is algebaric extension of $k$. (ii) Every irreducible polynomial over $k$, which has a root in $E$, has ...
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2answers
53 views

Automorphism of $\mathbb{C}$ over a subfield $K$ of $\mathbb{C}$

Assume a finite field extension $\mathbb{C}/K$ such that $[\mathbb{C}:K]>2$. Let $\varphi \in \text{Aut}(\mathbb{C}/K)$, so $\varphi \in \text{Aut}(\mathbb{C})$ and $\varphi\vert_K=\text{id}\vert_K$...
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35 views

Degree of $\mathbb{Q}(\sqrt{2 + \sqrt{7}})$ and splitting field

I have two questions: Determine the degree of the extension degree of $\mathbb{Q}(\sqrt{2 + \sqrt{7}})$ over $\Bbb Q$ and the degree of the splitting field of the minimal polynomial of $\sqrt{2 + \...
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1answer
23 views

Normal Closure of an Algebraic Extension

$\bf{Q.}$ Let $K/F$ be an algebraic extension. Show that there is an algebraic extension $L/K$ such that $L/F$ is normal and if $M$ is another normal extension of $F$ such that $F\subseteq K\subseteq ...
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3answers
60 views

$K=\Bbb Q(\sqrt3,\sqrt[3]{2}) $, Compute $[K:\Bbb Q]$.

I've tried looking at similar examples for but something has been close enough. How do I compute $[\mathbb Q(\sqrt 3,\sqrt[3]2):\mathbb Q]$. I think you say that it's equal to $[\mathbb Q(\sqrt3 + \...
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0answers
27 views

Quadratic Fields and Gauss Sums

I am given the p-cyclotomic field as $K_p = \frac{\mathbb{Q}[x]}{(\Phi_p(x))}$ where $\Phi_p(x) = x^{p-1} + x^{p-2} + ... + x + 1 $ Then, a quadratic subfield L is defined such that $\mathbb{Q} \...
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1answer
47 views

How is an arbitrary member of $\mathbb{Q}(\pi)$ defined?

How is arbitrary member of $\mathbb{Q}(\pi)$ defined? $\mathbb{Q}(\pi)$ means the extension of $\mathbb{Q}$ by $\pi$ , thanks I thought maybe it's $$x+y\pi,\quad x,y\in\mathbb{Q}$$ but how to ...
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2answers
25 views

Let $K$ be an extension of a field $F$ such that $[K:F] = 13$.Suppose $a$ $∈$ $K-F$.What is the value of $[F(a):F] $?

Since 13 is prime and we can write $[K:F]$ $=$ $[K:F(a)]$ $[F(a):F]$ , i think answer should be 1 or 13.
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A generator of a different form than the one given by the primitive element theorem

I want to find generators of the field extension $\mathbb{Q}(i,\sqrt[3]{2})$ over $\mathbb{Q}$. Using the primitive element theorem (see here), I get $pi+q\sqrt[3]{2}$, for every nonzero rational $p,q$...
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1answer
34 views

Let $L/K$ be a field extension and let $a, b \in L$ be algebraic elements over $K$ having the same minimal polynomial. Show that $K(a) \simeq K(b)$.

The above question was given to me as an assignment but I'm a bit stuck and I think the way I've been going about it is a dead end, or if not a dead end, I can't figure out a way to finish it off So ...
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1answer
20 views

Is the extension $\mathbb{Q}(\sqrt{5},\sqrt{7})$ simple? If so/not, why?

Exercise sounds: Is the extension $\mathbb{Q}(\sqrt{5},\sqrt{7})$ simple? If so/not, why?I have the solution (on picture). Is it correct? Why do we prove in this way, why we must show that square, ...
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3answers
69 views

Prove that $\Bbb Q(\sqrt{2},3^{1/3})=\Bbb Q(\sqrt{2}+3^{1/3})$

Prove that $\Bbb Q(\sqrt{2},3^{1/3})=\Bbb Q(\sqrt{2}+3^{1/3})$ My attempt: Firstly, since, $\Bbb Q(\sqrt{2}+3^{1/3}) \subseteq \Bbb Q(\sqrt{2},3^{1/3})$ , I computed $(\sqrt{2}+3^{1/3})^{-1} = 6-4\...
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1answer
24 views

If $u\in F$ is Transcendental over $K$, $F$ an extension field of $K$, Show every element in $K(u)$ not in $K$ is transcendental over $K$.

Doing some problems out of Beachy’s Algebra text, I came across that problem, and I’m at a loss how to show it without a bit of hand waving. Do I make some statement about spaces, and prove by ...
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1answer
79 views

Extension of Splitting Fields over An Arbitrary Field

Let $F$ be a field in which $0 \neq2$ in $F$, and consider $f=x^4+1$. If $E$ is the splitting field for $f$ over $F$, it turns out that $E$ is a simple extension of $F$. How does one realize this fact?...
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1answer
35 views

Kernel is a group of order $q+1$

Hello I hope you can help me with this doubt I have. Let $F_{q}$ a field and consider its quadratic extension $F_{q^{2}}=F_{q}(\sqrt{d})$ for $d$ square free in $F_{q}$ now we consider the next ...
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1answer
19 views

Series of Extension Fields

Let $ (a_n)_{n \in \mathbb{Z\geq0}} $ with $a_0 = 2 $ and $a_{n+1}= \sqrt{a_n}$ with $a_{n+1}>0$. We need to show that $[\mathbb{Q}(a_n):\mathbb{Q}]=2^n $$\forall n \in \mathbb{Q}$. To prove this ...
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Question about a Simple Field Extensions Equality

Let $E\supseteq F$ be an extension of fields. Show that $\forall u \in E,$ and nonzero $a\in F,$ $F(u)=F(au)$. My first instinct was to argue with the fact that $F(u)$ is the smallest subfield that ...
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1answer
51 views

Found $x^8$ while calculating inverse of $(x^6+1)$ in finite field $GF(2^8)$. Help???

So I was running the EEA (Extended Euclidean Algorithm) to find the multiplicative inverse of $(x^6+1)$ in the finite field $GF(2^8)$. Everything was going fine until the second last iteration where I ...
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2answers
64 views

Let $F$ be a field. Is it true that if $[F(\sqrt{D}) : F] = 2$, then $D \in F$?

In Dummit & Foote, problem 14.2.17(c), the authors hand us a quadratic extension of the form $F(\sqrt D)$. Now, while I am pretty sure you need that $D \in F$ to do this particular problem, I can'...
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0answers
42 views

Show that for any element α of some extension of F, E(α) is a splitting field of f over F(α)

Let F be a field and let f ∈ F[X] be a polynomial with a splitting field E over F. Show that for any element α of some extension of F, E(α) is a splitting field of f over F(α) I'm not really sure how ...
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0answers
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How to find the smallest extension field of $GF(p)$ which is a splitting field for all quotient groups $N_i/P_i$?

Let $F=GF(p)$ be the field with $p$ elements. Let $G$ be a finite group with order divisible by $p$. Let $S$ be a fixed Sylow $p$-subgroup of $G$ and let [$P_i$] be a list of representatives of $p$-...
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1answer
35 views

Complete valuation, norm of finite extension. Proof of Propositon.

I would like to ask for tips how to manage with proof of this proposition. How to show that $v$ can be uniquely extended to $v'$ ? Should I assume that $v$ can be also extended to another valuation? ...
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If $f_{K(\alpha)}^{\beta}\big| f_{K}^{\beta}$, then $\deg\left(f_{K(\alpha)}^{\beta}\right)| \deg\left(f_{K}^{\beta}\right)$?

Let $K\subset K(\alpha)\subset K(\alpha,\beta)$ be field extensions. Then the question is whether the minimal polynomials of $\beta$ over $K$ and $\beta$ over $K(\alpha)$ have degrees where one ...
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1answer
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Let $f(x)=x^4-x^2-1 \in \Bbb Q[x]$, $K$ is the splitting field of $f$ over $\Bbb Q$.

Let $f(x)=x^4-x^2-1 \in \Bbb Q[x]$, $K$ is the splitting field of $f$ over $\Bbb Q$. I’m to find out $\text{Gal}(K/\Bbb Q)$, how do I do? If the four roots of $f(x)$ in $\Bbb C$ are $x_1, x_2, x_3, ...
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1answer
35 views

Showing linear independence of power basis of $\mathbb{Q}(\sqrt[4]{3})$ over $\mathbb{Q}$.

I have been told that $\mathbb{Q}(\sqrt[4]{3})$ as a vector space over $\mathbb{Q}$ has dimension $4$. If $\alpha = \sqrt[4]{3}$, then I am guessing a basis is $1, \alpha, \alpha^2, \alpha^3$. I can ...
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1answer
40 views

Let $L/Q$ be a field extension. Let $\sigma\in\textrm{Aut}_Q(L)$. Let $f(x) \in Q[x]$ be a polynomial. Show that $f(σ(α)) = σ(f(α))$ for all $α ∈ L.$

Let $L/Q$ be a field extension. Let $\sigma\in\textrm{Aut}_Q(L)$. Let $f(x) \in Q[x]$ be a polynomial. Show that $f(σ(α)) = σ(f(α))$ for all $α ∈ L.$ The statement is obviously true for $α ∈ Q$ ...
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1answer
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Verifying a $K$-basis for a primitive extension $K\subset K(\alpha)$

Let $K\subset L:=K(\alpha)$ be a primitive field extension of degree $n$ and we define $c_i\in L$ as \begin{align*} \sum_{i=0}^{n-1}c_i x^i=\frac{f^{\alpha}_K}{(x-\alpha)}\in L[x]\quad(1) \end{...
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0answers
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Frobenius morphism associated to field extension

I've got a field extension $F_{q^n} / F_q$ of degree n and the Frobenius morphism $f$: $ x \to x^q$ associated to $F_{q^n} / F_q$. Let m be an integer dividing n. In terms of $f$, what's an ...
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1answer
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Questions about proof of existence of roots of $f$ in $K[X]/(f)$

Let $f \in K[X]$ with $deg(f)\geq 1$. Then there exists an algebraic field extension $L/K$, such that $f$ has a root in $L$. Proof: WLOG we can assume that $f \in K[X]$ is irreducible. Since $...
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4answers
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Is $\mathbb{Q}(\sqrt2, \sqrt3)$ isomorphic to $\mathbb{Q}(\sqrt[4]2)$?

I wish to show that $\mathbb{Q}(\sqrt2, \sqrt3)$ is not isomorphic to $\mathbb{Q}(\sqrt[4]2)$. What I did so far was to show that $\mathbb{Q}(\sqrt2,\sqrt3) =\text{span}\{1,\sqrt2,\sqrt3,\sqrt6\}$. ...
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2answers
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Proof that $F(a)=a+\langle f\rangle$ is an embedding from $K$ to $K[X]/\langle f\rangle$

I want to prove that when $F:K\rightarrow K[X]/\langle f\rangle $ is a map such that $F(a)=a+\langle f \rangle$, then $F$ is an embedding from $K$ to $K[X]/\langle f \rangle$, when $f\in K[X]\...
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$L_1L_2/K$ is separable. justify it

Is it true that- If $L_1/K$ and $L_2/K$ are extensions contained in a field $F$ and both are separable then $L_1L_2/K$ is separable. If not true then give me any counter example. Answer: In ...
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Why is $\mathbb{Q}(\zeta_7)$ not a radical extension?

My script states that there are $2$ non-trivial subfields of $\mathbb{Q}(\zeta_7)$, where $\zeta_7$ is a $7$-th primitive root of unity. These subfields are $\mathbb{Q}(\zeta_7 +\overline \zeta_7)$ ...
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1answer
33 views

Indeterminate is Algebraic over Transcendental Extension

I'm relatively new to the study of fields, and was presented with the following problem: Let $u = \frac{t^3}{t+1} \in K(t)$. Show $K(t)$ is algebraic over $K(u)$ and determine $[K(t):K(u)]$. I ...
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1answer
24 views

Finding a field extension in which every element has zero trace

Let F be a field extension of K, then F over K is a vector space, and for each a in F define f:F-->F as f(x)=ax, this is a linear transformation, define trace of a as trace of this linear ...
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Isomorphic field extensions have the same degree

Let $k_1 \subseteq k'_1$ and $k_2 \subseteq k_2'$ be field extensions. Suppose there is a field isomorphism $\phi: k'_1 \to k'_2$ where $\phi(k_1)=k_2$. Show that $[k_1':k_1]=[k'_2:k_2]$. Now my ...
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1answer
62 views

Can splitting field be generated by one root?

Say $f$ is an irreducible polynomial over a field $F$, and $\alpha$ is one of its roots, then is $F(\alpha)$ a splitting field for $f$? I tried to find some counterexample, but I failed.
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Are Valuations on algebraic extensions of an henselian field unique?

The definition of henselian is: A valued field $(\mathbb{K},\nu)$ is said to be Henselian if for any algebraic extension $\mathbb{L}$ of $\mathbb{K}$ there is a unique valuation $\tilde{\nu}$ on $\...
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16 views

Is $\mathbb Q(\sqrt 2) \times \mathbb Q(\sqrt 3)=\mathbb Q(\sqrt 2,\sqrt 3)$ if I prove $\sqrt 2,\sqrt 3$ are L.I. over $\mathbb Q$? [duplicate]

I proved that $\{1,\sqrt 2\}$ and $\{1,\sqrt 3\}$ are respective bases of $\mathbb Q(\sqrt 2)$ and $\mathbb Q(\sqrt 3)$ over $\mathbb Q$. I want to show in some sense that since $\sqrt 2,\sqrt 3$ are ...
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2answers
48 views

Why is $\mathbb{Z}_3[x]/\langle x^2+1\rangle$ a field with order $9$? And does order $9$ mean it has $9$ elements?

So I get that $\mathbb{Z}_3[x]/\langle x^2+1\rangle$ is a field since $x^2+1$ is irreducible. I don't get how to show it only has $9$ elements. And is that the same thing as saying it has order $9$?
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1answer
44 views

On the existence of an algebraically closed field containing other fields

This question arose while I was reading a paper I found in the web. It might be very simple, but I don't know the answer. Let $\mathbb{R}$ be the set of real numbers and $\mathbb{Q}_p$ the set of all $...
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1answer
16 views

About irreducible polynomial over field & characteristic or minimal polynomial of matrix

Let $F$ be a field and $K$ be a finite extension of $F$, and let $\alpha\in K$. Consider a linear map $T:K\to K$ is defined by $T(\beta)=\alpha\beta$ for all $\beta\in K$, where $K$ viewed as a ...
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2answers
41 views

Is $[\Bbb Q(5^{1/2}, 5^{1/7}): \Bbb Q(5^{1/7})]$ a normal extension?

I've working on a problem set with a bunch of these, and I get the idea generally. An extension is normal if all of the roots for the min pol for the element we are extending by are in the field. e.g.,...
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1answer
27 views

Is there an extension field of degree infinite has no intermediate field?

Studying a prime field, Prime fields has no proper subfields. At this point, i have some question. Suppose that $F$ is a field and $E$ is an extension of $F$. Suppose further that there is no ...
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1answer
15 views

Deducing if $\gcd(\deg(m_K(x)),\deg(m_K(y)))=1$ then $[K(x,y):K]=\deg(m_K(x))\times \deg(m_K(y))$.

I've shown that - If $x,y\in L$ are algebraic over $K$,then $[K(x,y):K]\le \deg(m_K(x))\times \deg(m_K(y))$. How can we deduce from the above result that if $$\gcd(\deg(m_K(x)), \deg(m_K(y)))=1,$$...
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0answers
10 views

prove that $K$ is the composite field of $K_i $'s

Let $\{f_i\}_{i \in I}$ be a family of polynomials in $F[X]$ (F is of course some field). Consider the extension field $K$ such that $f_i$ splits in $K[X]$ and is generated by all the roots of $f_i,i\...
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1answer
20 views

Determine all algebraic extensions of $\Bbb Q$ contained in $\Bbb Q(\sqrt{2},\pi)$

Determine all algebraic extensions of $\Bbb Q$ contained in $\Bbb Q(\sqrt{2},\pi)$ Assuming $\pi$ to be transcendental over $\Bbb Q$ , it seems to me that the answer must be only $\Bbb Q(\sqrt{2})$ . ...