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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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$\mathbb Q$-basis of $\mathbb Q(\sqrt[3] 7, \sqrt[5] 3)$.

Can someone explain how I can find such a basis ? I computed that the degree of $[\mathbb Q(\sqrt[3] 7, \sqrt[5] 3):\mathbb Q] = 15$. Does this help ?
user42761
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Let $\alpha$ be algebraic over $F$. Suppose the degree of $\alpha$ over $F$ is odd. Show that $F(\alpha)=F(\alpha^2)$?

I am trying to solve the following exercise: Let $\alpha$ be algebraic over $F$. Suppose the degree of $\alpha$ over $F$ is odd. Show that $F(\alpha)=F(\alpha^2)$ It's not clear to me how to solve this. I've been thinking and I guess that…
Red Banana
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Find $\gamma$ such that $\mathbb{Q}(\gamma)=\mathbb{Q}(\sqrt{-5},\sqrt{2})$

I have to solve the following problem: Find $\gamma$ such that $\mathbb{Q}(\gamma)=\mathbb{Q}(\sqrt{-5},\sqrt{2})$ I am a bit confused about how to do it but as $\mathbb{Q}(\sqrt{-5},\sqrt{2})$ has degree $4$,then $\mathbb{Q}(\gamma)$ must also…
Red Banana
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Is $\mathbb{Q}(i\sqrt[4]{2}) = \mathbb{Q}(1 + i\sqrt[4]{2})$?

Let $\alpha = i\sqrt[4]{2}$. Then, $1 + \alpha = 1 + i\sqrt[4]{2}$. Then, $\mathbb{Q}(\alpha) = \{ c_0 + c_1 \alpha + c_2 \alpha^2 + c_3 \alpha^3 \mid c_0, c_1, c_2, c_3 \in \mathbb{Q} \}$. and $\mathbb{Q}(1 + a) = \{ b_0 + b_1 (1 + \alpha) + b_2 (1…
xxxxxx
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$\mathbb{Q}(\pi^2-\pi) \subset\mathbb{Q}({\pi})??$

I have a "simple" question about field extension, let me say, $\mathbb{Q}(\pi^2-\pi) \subsetneq\mathbb{Q}({\pi})??$ I know that $\pi^2-\pi \in \mathbb{Q}(\pi)$ so we have $\mathbb{Q}(\pi^2-\pi) \subset\mathbb{Q}({\pi})$, but why $\pi \notin…
Joãonani
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If $\mathbb{K}$ is an extension of $\mathbb{Q}$ of degree $15$, which of these elements cannot belong to $\mathbb{K}$?

If $\mathbb{K}$ is an extension of $\mathbb{Q}$ of degree $15$, which of these elements cannot belong to $\mathbb{K}$ ? I know that it has to do with minimal polynomial but I am unsure of how to determine them.
user844532
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exercise 19.13 in Algebra (Issac)

Let $F\subset E$ be purely inseparable with $|E:F|=n<\infty$. If $\alpha \in E$, show that ${\alpha}^n \in F$.
Muniain
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How can we consider $\Bbb Z_3[x]/ \langle x^2+x+2\rangle$ as splitting field for $x^2+x+2$ over $\Bbb Z_3$

My book introduce two kind of splitting field. One of them is $\Bbb Z_3(i)=\{a+bi : a, b \in \Bbb Z_3\}$. The other is $\Bbb Z_3[x]/\langle x^2+x+2\rangle$. But, I am confused how the latter case could be splitting field. As follows my book, since…
fivestar
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Constructing a new monogenic extension of fields

Let $K / F$ and $L / F$ be two extension of fields, where $K / F$ is furthermore monogenic, such that $K$ and $L$ are isomorphic (as fields). Can I state that $L / F$ is monogenic too? If not, can I add extra conditions to $K$, $L$ and $F$ to get…
joseabp91
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Degree of a field extension is a power of 2

My question is somewhat similar to Dimension of a Finite Field Extension is a Power of 2 but there's a small complication. Claim 1: Let $K$ be a field of characteristic $\neq 2$, and suppose that every polynomial of odd degree in $K[x]$ has a root…
Joshua Ruiter
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E ∩ K = F implies [EK : F] = [E : F][K : F]?

I have come across a problem that shows E ∩ K = F if [EK : F] = [E : F][K : F]. I wonder if the converse holds. I know this reduce to showing the the F-basis of E is a K-basis if E ∩ K = F (if this is true), but is stuck here. Thanks.
assking
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Let $K$ be an extencion of $F$ and let $a,b\in K$. If $a$ is not algebraic in $F$, but it is in $F(b)$, show that $b$ is algebraic in $F(a)$

Let $K$ be an extencion of $F$ and let $a,b\in K$. If $a$ is not algebraic in $F$, but it is in $F(b)$, show that $b$ is algebraic in $F(a)$. $F(x)$ is the smaller subfield of $K$ containing $F$ and $x$. I'm not used to problem abouth field…
AlephZero
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Characterizing algebraic extensions

Show that $K$ is an algebraic extension over $F$ if and only if for every intermediate field $E$, every monomorphism $\sigma: E\rightarrow E$ which is the identity on $F$ is in fact an automorphism of $E$. The $\Rightarrow$ direction seems fairly…
Ducky
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Extension field of finite degree

Let $E$ and $F$ two finite extensions of a field $K$ of degree $[E:K]=m$ and $[F:K]=n$ such that $\gcd(m,n)=1$. Show that if $\alpha\in F$ has degree $r$ on $K$ therefore $\alpha$ has degree $r$ on $E$. What I did is, there is an irreducible…
idm
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What do you get when you add $i$ to the totally real numbers?

Say that an algebraic number $\alpha\in{\mathbb C}$ is totally real iff ${\mathbb Q}[\alpha]$ is a totally real number field. The totally real numbers obviously form a subfield of $\mathbb R$, which I will denote by $\mathbb T$. I ask myself…
Ewan Delanoy
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