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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.

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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Minimal polynomial with $f(\alpha)=0$

I'm learning this stuff for the first time so please bear with me. I'm given $\alpha=\sqrt{3}+\sqrt{2}i$ and asked to find the minimal polynomial in $\mathbb{Q}[x]$ which has $\alpha$ as a root. I'm ok with this part, quite sure I can find such a…
Ben Morine
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Show that $x^2+x+1$ is irreducible over $GF(2^n)$ for each odd $n$.

I am trying to show that $x^2+x+1$ is irreducible over $GF(2^n)$ for each odd $n$. Clearly it's true for $n=1$ since then $GF(2) = \mathbb{Z}_2$. I think the idea is to assume $u$ is a root of $x^2+x+1$ and then calculate $[GF(p^n)(u):GF(p^n)]$.…
fosho
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If $[K:k]=2$ then there is $a\in K$ s.t. $K=k(\alpha )$ and $\alpha ^2=a$.

1) Let $K/k$ an extension field. If $[K:k]=2$ show that there is $a\in K$ s.t. $K=k(\alpha )$ and $\alpha ^2=a$. 2) Does it work for $[K:k]=p$ when $p$ is prime ? Or for other number ? (like non-prime) My work For 1), if $[K:k]=2$ there is an $a\in…
MSE
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How to prove that $\mathbb{Q}(\sqrt[3]{2}+\sqrt{3})=\mathbb{Q}(\sqrt[3]{2},\sqrt{3})$?

It's clear that $\mathbb{Q}(\sqrt[3]{2}+\sqrt{3})\subseteq\mathbb{Q}(\sqrt[3]{2},\sqrt{3})$, but for the other direction, the most simple thing I could think of is to calculate powers of $\sqrt[3]{2}+\sqrt{3}$ (let's call it $\alpha$), and after…
asaelr
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tower of simple extensions of $\mathbb Z$$_p$ for which the full extension will not be simple

Let $s$ and $t$ be independent variables and let $p$ be a prime. Show that in the tower $\mathbb Z$$_p$$(s^p,t^p)\lt\mathbb Z$$_p$$(s,t^p)\lt\mathbb Z$$_p$$(s,t)$ each step is simple but the full extension is not. In fact it is easy to see that…
S.B.
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Is $\sqrt{2+\sqrt{5}}$ included in the extension field $\mathbb{Q}(\sqrt5)$?

Is $\sqrt{2+\sqrt{5}}$ included in the extension field $\mathbb{Q}(\sqrt5)$? The motivation for this question is the understanding that the square root of a Complex Number $(\sqrt{a_0+b_0i})$ is itself a Complex Number, $(a_1+b_1i)$. (I know. I'm…
MathAdam
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If $\alpha^{2}$ is algebraic over F, then show that $\alpha$ is algebraic over F.

I have that $\exists p(x) \in F[x]$ with $p(x)=0$ then $p(a^2)=0$ if I take $p(x):=b_0+b_1x+b_2x^2+...+b_nx^n$ then $p(a^2):=b_0+b_1a^2+b_2a^4+...+b_na^{2n}=0$. I want to prove that $\exists q(x) \in F[x]$ with $q(x)=0$ so $q(a)=0$ if I take…
Estif Emmanuel Rodriguez
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Degree of Extension from $\mathbb{Q}$ to $\mathbb{Q}(i, \sqrt[4]2)$

I wanted to find the degree of extension from $\mathbb{Q}$ to $\mathbb{Q}(i, \sqrt[4]2)$. The minimal polynomial of $\sqrt[4]2$ over $\mathbb{Q}$ is $f(x)=x^4-2$. Now the roots of $f(x)$ are $\{\sqrt[4]2, -\sqrt[4]2, \sqrt[4]2i, -\sqrt[4]2i\}$. So…
Josh
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$f(x)=x^3-21x+35$ is irreducable over $Q$. Show if $f(\alpha)=0$ then $f(\alpha^2+2\alpha-14)=0$

I am not sure how to do this in a short, nice way. I have proved it by factoring $f(x)=(x-\alpha)$ and plugging in, but that is very very messy. Is there a slick way? Thank you!
Sorfosh
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If $\sqrt[3]{2}+\omega\in \Bbb{K}$, prove that $\sqrt[3]{2}\in \Bbb{K}$

Let $K=\Bbb{Q}(\sqrt[3]{2}+\omega)$ be a field extension of $\Bbb{Q}$ where $\omega$ is a root of $x^2+x+1$. Prove that $\sqrt[3]{2}\in K$ How do I prove this? I am at a loss. I did it by a very long method, involving solving for coefficients. I'm…
user67803
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Simple extension of a countable field is also countable?

How do we know "Simple extension of a countable field is also countable?" This intuitively makes sense but I'm not sure how to rigorously prove this. I want to use this to prove that there does not exist a simple extension from $\mathbb{Q}$ to…
nekodesu
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Compositum of normal extensions and embedding

I want to show "If $E_1$ and $E_2$ are normal extensions of $F$, then the compositum $E_1 E_2$ and $E_1\cap E_2$ are normal extensions." I already prove this, but I find another proof using an embedding $\sigma$ on $\overline{F}$(not…
Yongha
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Check whether roots exist for a polynomial is splitting field of another polynomial

Given that $F$ is the splitting field over $\mathbb{Q}$ of $x^{13}-1$. If I have another polynomial, say $x^9-3x^5+7x+10$ , how can I say whether it will have roots in $F$ or not? I know that for $x^{13}-1$, $F$ will be equal to $\mathbb{Q}\left(1,…
Gops76
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Why $\mathbb Q(i, \sqrt{-2}) : \mathbb Q$ is an extension of degree 4

Clearly $[\mathbb Q(i): \mathbb Q] = 2$. I wanted to prove that $X^2+2$ is irreducible over $\mathbb Q(i)$ in order to get the degree of the whole extension. My attempt: Assume $X^2+2$ is reducible. Than there must be an $\alpha \in \mathbb Q(i)$…
user42761
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Splitting field of a polynomial of degree 6

I am convinced that the splitting field over ${\mathbb Q}$ of $f(t)=t^6-10t^3+22$ has degree 36. This is equivalent to prove that $\sqrt[3]{5+\sqrt{3}}$ does not belong to ${\mathbb Q}(\sqrt[3]{5-\sqrt{3}})$ and I am not able to check this. Could…
JOSE MANUEL GAMBOA MUTUBERRIA
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