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Mathematics Stack Exchange

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.

An extension field $K$ of the field $F$ (denoted $K/F$) is a field such that $F$ is a subfield of $K$.

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I'm wondering definition of "E is a splitting over F"

I know the definition of splitting field $E$ for $f(x)\in F[x]$ but I am confused by this definition: $E$ is a spliiting over $F$ (not $f(x)$). I want to know that... please answer!
hobin
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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 $\mathbb{C}$ such that $\sigma (K)=K$ and $\sigma…
MAS
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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 $\Bbb Q(\sqrt{2})$ . $\pi$ transcendental $\implies {\pi}^n $…
user422112
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The degree extension $Q(\sqrt{2},\sqrt[3]{2},\sqrt[4]{2})$

So as in the title i am looking for the extension $Q(\sqrt{2},\sqrt[3]{2},\sqrt[4]{2})$. First of all i know that $[Q(\sqrt[3]{2}):Q]=3$ as $x^3-3$ is irreducable and so is the minimal polynomial. I also know that…
Sorfosh
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Field not separable

Are there fields which are not separable? I have seen that the field of characteristic 0 and finite fields are all separable or has separable extensions.
James Reid
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Degree of Field Extension of Number Fields

I'm trying to find the degree of the Field extension $\mathbb{Q}(\sqrt{-1},\sqrt{3}) $ over $ \mathbb{Q}(\sqrt{3}) $, namely $$ \vert\mathbb{Q}(\sqrt{-1},\sqrt{3}): \mathbb{Q}(\sqrt{3}) \vert = n $$ Does one usually approach such a problem with the…
Schief
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degree of [$\mathbb Z_2[x]/f\mathbb Z_2[x]: \mathbb Z_2$]

I got maybe easy problem. I am not sure if it is true that [$\mathbb Z_2[x]/f\mathbb Z_2[x]: \mathbb Z_2$]=deg $f$ where $f \in \mathbb Z_2[x]$ irreducible. Can anybody help me ? Thanks
mrbean123
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How to describe the field extension?

Consider the circle with radius $2\sqrt{2}$ centered at the origin and the line joining the points $\left(\frac{1}{2}, 0\right)$, $(4\sqrt{2}, \sqrt{2})$. The points of intersection of the line and circle have coordinates which lie in a field…
user326752
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Proving that having a unique embedding implies purely inseparable.

In the problem statement, we are given $k \subset K$ an algebraic field extension of characteristic p > 0, L an algebraically closed field containing K, and $\delta: k \rightarrow L$ be an embedding. What I'm trying to show is that if there exists…
user31415926
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If $K\subset F_i\subset L$, $i=1,2$ are two normal extension of $K$, show that $F_1\cap F_2$ is normal on $K$.

If $K\subset F_i\subset L$, $i=1,2$ are two normal extensions of $K$, show that $F_1\cap F_2$ is normal on $K$. I have absolutely now idea how to proceed. First, is a normal extension an algebraic extension ?
Rick
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Verify that $\sqrt{1+\sqrt2+\sqrt3+\sqrt5}$ is constructible by determining the sequence of its extension fields

Verify that the following numbers are constructible by determining the sequence of their extension fields: $$ \sqrt{1+\sqrt2+\sqrt3+\sqrt5},…
hohner
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How many extension dimension of Q field

How many extension dimension of $\Bbb Q(\omega_3,\omega_5)$, $\Bbb Q(2^{1/3},i)$and $\Bbb Q(2^{1/3},\omega_3)$ where $\omega_n=e^{2\pi i/n}$ the first i think 6 extension the second is three third is 4
user146264
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What is $[\mathbb{Q}[\sqrt[4]{2},\sqrt{-1}]:\mathbb{Q}]$?

What is $[\mathbb{Q}[\sqrt[4]{2},\sqrt{-1}]:\mathbb{Q}]$ ? Is it just $4\cdot2=8$? Is $\sqrt[4]{2} \in \mathbb{Q}[\sqrt{-1}]$?
LOIS
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Field Extensions - Algebraic Extensions

I have a question, How can I prove that $[\mathbb{Q}({\sqrt{2},\sqrt{5},\sqrt{10}}):\mathbb{Q}]= 4$?. Thank you.
Halsey
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Extension of fields - Number of elements

If K is an n-dimensional extension field of $Z_p$, what is the maximum possible number of elements in K? I am a biginner and i dont understand all the procedure to proof that. I dont know how to start it. Thanks for the help!!
Timóteo A. Cruz
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