# 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.

3,134 questions
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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $K[x+y] \subseteq K[x, y]/(xy-1)$ 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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### 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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### 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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### 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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### 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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### 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 $k$. ...
### 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,...
### 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 ...