# 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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### The Relationship Between Field Extension Tower and Tensor Product

Let $F$ be a field. Consider field extension $F \subset L \subset K$. Let $\alpha_1, \dots , \alpha_m$ be an $F$-basis of $L$, $\beta_1, \dots, \beta_n$ an $L$-basis of $K$. By the theory of ...
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### Inclusion of field extensions $\mathbb{Q}(\omega_p)$ and $\mathbb{Q}(\omega_{2p})$

Could you help me clear up a confusion? Either my following reasoning is wrong, or I should be able to show that $\omega_{2p} \in \mathbb{Q}(\omega_p)$, which does not seem correct. Degree of Field ...
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### Is there a separable extension of degree 21?

Given the field $F= \mathbb{F}_3$ and a transcendental $t$, I am trying to find an intermediate field $$F(t^{1/63}) \supset E \supset F(t)$$ where $[F(t^{1/63}): E]= 21$ and the extension is ...
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### Computing $[\mathbb{C}(x,y):\mathbb{C}(xy^n+x^ny,xy)]$ and similar extensions

The following three questions are similar to this question. For every natural numbers $k,l,m,n \geq 2$, define: $A_{n}=\mathbb{C}(xy^n+x^ny,xy)$. $B_{m,n}=\mathbb{C}(xy^m+x^my,xy^n+x^ny)$, in ...
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### What is the Galois Group of $(x^8 - 1)(x^2 + 2)$ over $\mathbb{Q}$?

I am studying for some algebra qualifying exams over the summer and I am stumped on the title question: What's the Galois Group for $(x^8 - 1)(x^2 + 2)$ over $\mathbb{Q}$? Here's what I've got so far: ...
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### Problem (9.19) - Character theory of finite groups (Martin Isaacs)

I am currently reading chapters $9$, $10$ et $15$ of Isaacs' character theory of finite groups and I am stuck with this question on splitting field. In this case, I have to found a splitting field for ...
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### Proof of the existence of algebraic closure

I am taking an introductory course on Galois theory, and we recently covered the following theorem: For every field $\mathbb{F}$, there exists a field extension $\mathbb{L}$ which is algebraically ...
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### Show that $\mathbb{Q}(\sqrt[3]{5} \cdot \sqrt{2}) \subset \mathbb{Q}(\sqrt[3]{5}+ \sqrt{2})$

How can I show that the extension $\mathbb{Q}(\sqrt[3]{5} \cdot \sqrt{2})$ is contained in the extension $\mathbb{Q}(\sqrt[3]{5} + \sqrt{2})$? I need to show that $\sqrt[3]{5} \cdot \sqrt{2}$ can be ...
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### Let $E$ be an extension of $F$ and let $A = \{e \in E: e$ is algebraic over $F\}$. Show that $A$ is a subfield of $E$ containing $F$.

Let $E$ be an extension of $F$ and let $A = \{e \in E : e$ is algebraic over $F \}$. Show that $A$ is a subfield of $E$ containing $F$. First, clearly $A \subseteq E$. Also since $F$ is algebraic over ...
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### Why is the subfield of $\mathbb{Q}(\zeta_p)$ of index $2$ expressible in terms of the sum of $\zeta_p$ to the power of all quadratic residues mod $p$?
Let $p$ be an odd prime, $\zeta_p = e^{2 \pi i / p}$. I've been playing around with calculating the intermediate fields of $\mathbb{Q}(\zeta_p)$. I know that \$\mathrm{Gal}(\mathbb{Q}(\zeta_p) / \...