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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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23 views

Proving that a field extension is Galois

Okay, for an assignment I'm seeking to show that a field extension is Galois. However we never really went into detail on proving such things, at least with concrete examples, and I'm having trouble ...
2
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1answer
28 views

If $[K:\Bbb{Q}]=2$ then $K=\Bbb{Q}(\sqrt{d})$.

I am stuck on one question and sincerely have no idea how to proceed. Let $K$ be a field containing $\Bbb{Q}$ such that $[K : \Bbb{Q} ] = 2$. Prove that there exists a square free integer $d$ such ...
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1answer
38 views

A question on the Galois correspondance of $\sigma^2$ in $D_8$.

In the context of Galois theory the splitting field of $x^4-3$ is isomorphic to $D_8$. Therefore one of the elements of this group is $\sigma$, which maps $i\rightarrow i$ and $\sqrt[4]{3}\rightarrow ...
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0answers
22 views

Root of a nonzero polynomial with algebraic coefficients is algebraic? [duplicate]

I was solving a question and one of the step in that question requires above statement .While studying abstract algebra i didn't studied anything like root of a nonzero polynomial with algebraic ...
3
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2answers
35 views

Find degree of extension $Q$($\sqrt{1+\sqrt{-3}}$ + $\sqrt{1-\sqrt{-3}}$) over $Q$

I tried solving this textbook problem.Any hint how to simplify or find the degree of extension in this case ?I guess maximum degree can be 4.
2
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1answer
55 views

If $x$ and $y$ are complex numbers and $x+y$ , $xy$ are algebraic numbers then how to prove that $x$ and $y$ are also algebraic numbers?

I tries basic operations like multiplication and addition in a hope that i will get $x$ and $y$ out of $x+y$ and $xy$ but that didn't worked for me.Also i tried assuming a polynomial with rational ...
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0answers
5 views

If $k(x,y) = \sum_r^n(\prod_{k\not =r}^n g_k(x) )*(f_r(x)) y^r$ is nonzero in the powers of $y$, is it nonzero in the powers of x also?

Let $u,v \in F/K$, a field extension of $K$ s.t $v$ is transcendental over $K$, and assume we know that for $$q(x) = \sum_r^n (\prod_{k\not =r}^n g_k(u) )* (f_r(u) x^r \in K(u)[x],$$ $q(v) = 0$, where ...
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1answer
23 views

Prove that $[E(u): F(u)] \leq [E:F]$.

Given tower of fields $K\supseteq E\supseteq F $, prove that for $u\in K$ algebraic over $F$ and $[E:F]$ finite. This is a problem from W. Keith Nicholson's book. My idea is that $[E:F]$ is finite ...
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1answer
38 views

When degree of splitting field equals n factorial

Given that the degree of splitting field of a polynomial $f(x)$ over $\mathbb{Q}[x]$ is equal to $n!$ where $n$ is the degree of $f$, $n>2$. If $\alpha$ is a root of $f$ in the splitting field, ...
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39 views

For a field $K$ of characteristics $p>0$, when is a finite purely inseparable extension $F/K$ (with $[F:K]=p^n>1$) such that $F\cong K$?

An example that illustrates the question is: $F=\mathbb{F}_p(t)$ and $K=\mathbb{F}_p(t^p)$, for which $F\cong K$ by Luroth's theorem. Also, for $p>3$, consider $F=\overline{\mathbb{F}}_p(x,y)...
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+50

What is the Galois group of the Hyperreal Numbers?

The Galois Group of $\mathbb{R}$ as an extension of $\mathbb{Q}$ is trivial. That’s because any field automorphism of $\mathbb{R}$ is order preserving, so since $\mathbb{Q}$ is dense in $\mathbb{R}$, ...
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2answers
69 views

Factorization of a polynomial in $\Bbb F_7$

I need to reduce as much as possible the polynomial: $x^6+3x^5+2x^4+6x^3+4x^2+5x+2$ in the finite field $\Bbb F_7$. It has no roots over the field, and I don't think that it is necessary to check ...
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1answer
29 views

Let $F = \mathbb{Q}(a_1, a_2, . . . , a_n)$ with $a_i^2 \in \mathbb{Q}.$ Prove that $ \sqrt[3]2 \notin F$.

So I tried by claiming each extension $ \mathbb{Q}(a_i) $ was of degree 2 because of the $a_i^2 \in \mathbb{Q}.$ Apparently that wasn't necessarily true. I said that $ \mathbb{Q}(\sqrt[3]2) $ was a ...
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0answers
24 views

Tower of Splitting Fields

If we have $F_n\geq\ldots\geq F_1$ fields such that $F_{i+1}$ is a separable splitting field over $F_i$ for all $i=1,\ldots,n-1$ is it true that $F_n$ is a splitting field over $F_1$? I think I can ...
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35 views

Ramification Examples

Any help for seeking a totally tamely ramified extension of $\mathbb{F}_3$((X)), an unramified extension of $\mathbb{F}_3$((X)), a totally tamely ramified extension of $\mathbb{Q}_3$(i)? Feel ...
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0answers
26 views

List of extension theorems

As a post grad student, I have come across many results where a function with certain properties(eg-homomorphism) on a smaller algebraic structure is extended to a larger one. For example, extending ...
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1answer
62 views

Show that there exists an $\alpha \in K$ such that $\alpha^2 \in F$ but $\alpha \notin F$

Let G be a group of order $2^n$ and suppose that $G=Gal(K/F)$ where $F \subseteq K$ is a Galois, separable, normal extension. Then show that there exists an $\alpha \in K$ such that $\alpha^2 \in ...
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2answers
57 views

minimal polynomial for $\sqrt{-3}+\sqrt{2}$ over $\mathbb{Q}$

Almost have the answer. Let $a = \sqrt{-3}+\sqrt{2} \implies (\sqrt{-3}+\sqrt{2})(\sqrt{2}-\sqrt{-3}) = 5 \therefore $ a is a root of $x^4+2x^2+25$. $\sqrt{2} = \frac{1}{2}(a+5/a) \in \mathbb{Q}(a), \...
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1answer
16 views

How to find the basis of an extension field

Sorry for asking a simple question but why is it obvious that $\{1,\sqrt{3}+\sqrt{5}\}$ is a basis of $\mathbb{Q}(\sqrt{3}+\sqrt{5})$ over $\mathbb{Q}(\sqrt{15})$? I know that $[\mathbb{Q}(\sqrt{3}+\...
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2answers
167 views

Simplicity of the roots of a minimal polynomial

Let $L/K$ be a finite field extension, and let $\mu_{\alpha,K}\in K[X]$ be the minimal polynomial of $\alpha\in L$. One can easily see that $\alpha$ is a simple root of $\mu_{\alpha,K}$. Indeed, if ...
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2answers
40 views

$\mathbb{Q}(\alpha)\cap\mathbb{R}=\mathbb{Q}$, where $\alpha=\sqrt{\frac{3+\sqrt{7}i}{2}}$

How can I show $\mathbb{Q}(\alpha)\cap\mathbb{R}=\mathbb{Q}$ where $\alpha=\sqrt{\frac{3+\sqrt{7}i}{2}}$? $\alpha$ is a root of a degree 4 irreducible polynomial over $\mathbb{Q}$, so $\mathbb{Q}(\...
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0answers
52 views

Field Extension $ \mathbb{Q} \subset \mathbb{Q}_p $ infinite

Let $\mathbb{Q}_p$ be the $p$-adic rational field. I want to verify that the field extension $\mathbb{Q}_p/ \mathbb{Q}$ is infinite therefore $\dim_{\mathbb{Q}}(\mathbb{Q}_p) = \infty$. My ...
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1answer
12 views

If $\phi$ is an F-map from $K$ to $E$, both field extensions of $F$, then $\alpha \in K$ and $\phi (\alpha)$ have the same minimum polynomial

Definition of an F-map: If $K$ and $E$ are field extensions of $F$, an F-map is a homomorphism, $\phi: K \rightarrow E$ such that $F$ is fixed. I'm reading over a proof of why the number of ...
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1answer
27 views

A question regarding finding the minimal polynomial associated with a field extension . [duplicate]

Say we have the field extension $\Bbb Q(w,\sqrt[3]{5})$ over $\Bbb Q$, where w is the primitive cubed root of unity. I know that the minimum polynomial of $\sqrt[3]{5}$ is $x^3-5$. I want to figure ...
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3answers
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Finding degree of a finite field extension

Let $x=\sqrt{2}+\sqrt{3}+\ldots+\sqrt{n}, n\geq 2$. I want to show that $[\mathbb{Q}(x):\mathbb{Q}]=2^{\phi(n)}$, where $\phi$ is Euler's totient function. I know that if $p_1,\ldots,p_n$ are ...
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0answers
23 views

Homomorphism of splitting field to its closure

Let $k$ be a field, $f(x)\in k[x]$, and let $F$ be the splitting field of $f(x)$ over $k$. Let $k\subseteq K$ be an extension such that $f(x)$ splits as a product of linear factors over $K$. Prove ...
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0answers
28 views

Why is $GF(p^n)$ unique? [duplicate]

I'm having trouble understanding why exactly $GF(p^n)$ is unique up to isomorphism. I know that the proof begins by claiming that $GF(p^n)$ is the splitting field of the polynomial $x^{p^n}-x\in \...
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0answers
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extension of field homomorphism

Let $K\subset L\subset M$, where $L$ is an algebraic field extension of $K$, and $M$ is an algebraic field extension of $L$. Consider a $K$-homomorphism $\phi\colon L\to\overline K$, where $\overline ...
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1answer
55 views

If $f \in F[x]$ irreducible, ${\rm char}(F) = p$, then $f(x) = g(x^{p^e})$ and every root of $f$ has multiplicity $p^e$ in some splitting field

Let $f(x)$ be irreducible in $F[x]$, $F$ of characteristic $p>0$. Show that $f(x)$ can be written as $g(x^{p^e})$ where $g(x)$ is irreducible and separable. Use this to show that every root of $f(x)...
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1answer
29 views

$K[x+y] \subseteq K[x, y]/(xy-1)$ is an integral extension

Let $K$ be a field. Prove that $K[x+y] \subseteq K[x, y]/(xy-1)$ is an integral extension. I know that $K[x,y]/(xy-1) \simeq K[t, t^{-1}]$, but I'm not sure if this would be useful to prove the ...
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1answer
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Matrices similarity in a bigger field $K$ Implies matrices similarity in the smaller field $F$.

I have a Linear Algebra exercise and I have trouble solving a part of it. The follwing question shows us that if $K \subseteq L$ is a field extension such that both $L,K$ are infinite ($L,K$ are ...
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1answer
63 views

Galois group of a quintic with 3 real roots. How to conclude that there's one cycle of order 5?

I understand perfectly the argument making use of Cauchy's theorem, which I'll lay down for clarity's sake: take $p(x)$ of degree 5 irreducible over $\mathbb{Q}$. Let $K$ be the root field of $p(x)$ ...
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1answer
34 views

$\Bbb Q(\sqrt{D_f})$ is an intermediate field with degree of extension $2$

Let $K |\Bbb Q$ be a cyclic extension of even degree.Let $f \in \Bbb Q[X]$ be an irreducible polynomial of degree $[K:\Bbb Q]$ having a root in $K$ .Then show that the unique field $F$ such that $\Bbb ...
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0answers
19 views

Field extension similarity

Let $F \subseteq K$ Be a field extension.I am Trying to prove that if $A,B \in M_n(F)$ are similar as matrices over the field $K$ (there exists an invertible matrix $P \in M_n(K)$ such that $PA=BP$) ...
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0answers
16 views

Every embedding of a splitting field into its algebraic closure, is an automorphism

I would like verification of my proof of the above claim. That is, I am trying to prove that if $K\subseteq\overline{F}$ is a splitting field for $\{f_i\}_I$ over $F,$ where $\overline{F}$ is an ...
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0answers
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If $k_1, \ldots, k_n$ are non-square, pairwise coprime, then $\sqrt {k_n} \not \in \mathbf{Q}(\sqrt {k_1}, \ldots, \sqrt {k_{n-1}})$ [duplicate]

Seems intuitive. Like the fact that $\sqrt 3 \not \in \mathbf{Q}(\sqrt 2)$. But how to approach actually proving it? The proof of this fact doesn't seem to generalize well.
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extending automorphisms in a tower of fields

Say we have the following tower of fields: $\mathbb{Q} \subset \mathbb{Q}(\sqrt{5}) \subset \mathbb{Q}(\sqrt[4]{5})$. For ease, we can write $\mathbb{Q} = F, \mathbb{Q}(\sqrt{5}) = E, \mathbb{Q}(\sqrt[...
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2answers
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A question about extension field

Example from Gallian: Let $f(x) = x^5+2x^2+2x+2\in Z_3[x]$. Then the irreducible factorization of $f(x)$ over $Z_3$ is $(x^2+1)(x^3+2x+2)$. So to find an find an extension $E$ of $Z_3$, in which $f(x)$...
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1answer
55 views

Prove that the Galois Group Is $D_{10}$

I've been working on this problem for a couple days and I haven't heard back from any of my support team (TAs/Prof) at Uni. Not looking for an answer, but help and hints would be greatly appreciated. ...
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0answers
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Let $K=\mathbb{Q}(\alpha \xi)$. Then choose the correct statements

Let $\alpha=\sqrt[5]{2} \in \mathbb{R}$ and $\xi=e^{\large \frac{2 \pi i}{5}}$. Let $K=\mathbb{Q}(\alpha \xi)$. Then choose the correct statements: $(i)$ There exists a field automorphism $\sigma$ of ...
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1answer
42 views

Intersection of Fields

I have a question about an argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (see page 124): My question is why does $L_0 \cap K \bar{k}=KL$ hold. One inclusion ...
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1answer
22 views

coefficient extension for fraction field $K(T) \otimes_K L$

let $L/K$ be an algebraic field extension. denote by $K(T)= Frac(K[T])$ the transcendental field extension of $K$. I would like to find out how to show that the equation $$K(T) \otimes_K L = L(T)$$ ...
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2answers
122 views

With $[K: F] = 3, \alpha \in K, \beta \in K \backslash F$, show that there are $a,b,c,d \in F$ such that $\alpha = \frac{a+b\beta}{c+d\beta}$

The original question states: Let $K$ be an extension of $F$, $[K:F] = 3$. Show that for every $\alpha \in K, \beta \in K \backslash F$, $\alpha = \frac{a+b\beta}{c+d\beta}$ for some $a,b,c,d \in ...
0
votes
1answer
24 views

$[L:K]$ dividing $n!$ with $L=K(\alpha_1,…\alpha_n)$, and $L$ a splitting field of $f$ with $\alpha_1..\alpha_n$ zeros of $f$

I have just proven the following statement: Let $K$ be a field and $L$ the splitting field of a separable polynomial $f\in K[X]$ of degree $n$. Denote the zeros of $f$ in $L$ by $\alpha_1,\alpha_2,......
2
votes
1answer
32 views

Let $k \subseteq F \subseteq K$ be fields, and let $z \in K$. Prove that if $k(z) \colon k$ is finite, then $[F(z):F] \leq [k(z):k]$.

Let $k \subseteq F \subseteq K$ be fields, and let $z \in K$. Prove that if $k(z) \colon k$ is finite, then $[F(z):F] \leq [k(z):k]$. In particular, $[F(z):F]$ is finite. If $k(z) \colon k$ is finite,...
1
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1answer
48 views

Tensor Product over Algebraically Closed Field

I have a question about a statement/formulation in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (see page 122): We fix an integral proper normal curve $X$ over a field $k$. ...
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0answers
48 views

Tensor Product of Fields is a Field

I have two questions about a construction introduced in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (see page 122): We fix an integral proper normal curve $X$ over a field ...
2
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2answers
51 views

Show that $\mathbb{Q}(\sqrt{1+\sqrt{3}})\cap \mathbb{Q}(\sqrt{1-\sqrt{3}})=\mathbb{Q}(\sqrt{3})$.

Show that $\mathbb{Q}(\sqrt{1+\sqrt{3}})\cap \mathbb{Q}(\sqrt{1-\sqrt{3}})=\mathbb{Q}(\sqrt{3})$. I know that by closure, $\sqrt{3}\in \mathbb{Q}(\sqrt{1+\sqrt{3}})\cap \mathbb{Q}(\sqrt{1-\sqrt{3}})$,...
3
votes
3answers
109 views

How to show that $\sqrt{2}$ is not in $\mathbb Q(\sqrt{3},\sqrt{5})$?

How to show that $\sqrt{2}$ is not in $\mathbb Q(\sqrt{3},\sqrt{5})$? First I tried to use the theorem that if $b$ is in $F(a)$, then $\deg(b,F)$ divides $\deg(a,F)$. But the theorem can not be ...
1
vote
1answer
23 views

Field extensions contradicting tower law

$[\mathbb{R}(x):\mathbb{R}(x+\frac{1}{x})]=2$ since $x$ satisfies a quadratic polynomial over $\mathbb{R}(x+\frac{1}{x})$. Similarly, $[\mathbb{R}(x):\mathbb{R}(x^2+\frac{1}{x^2})]=4$. But, as $(x+\...