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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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dimension of $\mathbb{C}$ over $\mathbb{Q}$

I am studying about extension fields.I want to know what would be [$\mathbb{C}$ : $\mathbb{Q}$] ?
kp9393
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Constructing Field Extensions from Irreducible Polynomials

Suppose f(x) $\in$ F[x] is a non-zero irreducible polynomial of degree n < $\infty$ over a field F, then E = F[x]/(f) is a field extension E\F of degree n. My question is: Can every field extension of F that is of finite degree be constructed in…
mattapow
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Prove that for all $n \ge 2$ there exist finite field extensions of $\mathbb{Q}$ of dimension $n$.

Intuitively, for every $n \ge 2$ I can find an element $u=a^{1\over{n}}$with $a \in \mathbb{Q}$ and an irreducible polynomial in $\mathbb{Q}[X]$: $u^{n}-a=0$. As $u \notin \mathbb{Q}$, then $[\mathbb{Q}(u)/\mathbb{Q}]=n$ which is a finite…
jjjx
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Recognising to be a field

I am sorry if this is too well-known. Is $\mathbb{Z}[\omega]$ a field ? Here $\omega$ is a non real $p^{\text{th}}$ root of $1$ and $p$ is a prime. And prove whatever your claim is. On an unrelated topic I was curious to know that to show…
shadow10
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Dimensions in field extensions

How would I be able to determine $[\Bbb Q(\sqrt{42}, \sqrt{-42}):\Bbb Q]$? So far, I think the dimension might be four as the root equation could be $(x^2 + 42)(x^2 - 42)$.
user126731
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field extension with degree of 2

I need to decide if this following statement is true or false: $\forall K \text{ field (not algebraically closed) } \exists L,K\leq L: [L:K]=2$ For finite fields this statement should be true, but I have no idea if this is true for all $K$.
mast
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powers in field extension

If $(3+ \sqrt2 )^n +r^n$ is always integer, then there exists $r=3-\sqrt2$ satisfying the equation. Then, given $(1+\sqrt2 +\sqrt3 +\sqrt6 )^n +x_1^n+x_2^n +...+x_m^n$ is an integer for all natural numbers $n$, the smallest natural number $m$ such…
추민서
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F-Homomorphism between two field extensions

If $K$ and $L$ are two extensions of a field $F$ then they are said to be $F$-isomorphic if there exists an isomorphism $\phi : K\rightarrow L$ which when restricted to the subfield $F$ forms an identity map. Similarly I assume that…
Siddharth Prakash
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Is that $[\mathbb{C} : \mathbb{R}] = 2$ and is that $[\mathbb{C} : \mathbb{Q}]$ infinite?

Since, $\mathbb{C} = \mathbb{R}(i)$ where $i = \sqrt{-1}$. So, $[\mathbb{C} : \mathbb{R}] = [\mathbb{R}(i) : \mathbb{R}] = 2$. Which $\mathbb{C}$ is a finite field extention of $\mathbb{R}$. Since, there are infinite number of algebric numbers in…
xxxxxx
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If $\beta$ is transcendental over $F$, then is that $F(\beta)$ must be an infinite extention of $F$?

Let $F$ be fields. If $\beta$ is transcendental over $F$, then is that $F(\beta)$ an infinite extention of $F$ ? Or $F(\beta)$ can be an finite extention of $F$ ?
xxxxxx
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How can I show this field extension equality?

How can I show this field extension equality $\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \mathbb{Q}(\sqrt{2} + \sqrt{3})$?
Kenan123
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$L/k$ and $K/k$ field extensions of $k$ with $[L:k]=m,[K:k]=n, (m,n)=1\implies L\cap K=k$

$L/k$ and $K/k$ field extensions of $k$ with $[L:k]=m,[K:k]=n, (m,n)=1\implies L\cap K=k$. The first thing I taught is that if $\alpha\in L,\beta\in K$ are algebric over $k$, the degrees of the minimal polynomials must not divide each other, so the…
Mateus Rocha
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extension of field homomorphism

Let $K\subset L\subset M$, where $L$ is an algebraic field extension of $K$, and $M$ is an algebraic field extension of $L$. Consider a $K$-homomorphism $\phi\colon L\to\overline K$, where $\overline K$ is an algebraic closure of $K$. I want to show…
Sha Vuklia
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Extending an automorphism from $\mathbb{Q}(\sqrt{2})$ to an automorphism of $\mathbb{Q}(\sqrt[4]{2})$

I am reading an introduction to abstract algebra by Keith Nicholson. There is a theorem that goes like this : Let $\sigma : F \rightarrow{\overline{F}}$ be an isomorphism of fields, let $f \in F[x]$ be a nonconstant polynomial, and let …
roi_saumon
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The sum of two algebraic points is a new algebraic point

Let $K / F$ be an extension of fields and let $x , y \in K$ be two algebraic points over $F$. I do not know if my proof of $x + y$ is an algebraic point over $F$ is correct. My attempt is the next: checking that $F(x , y) / F$ is a finite extension…
joseabp91
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