Mathematics Stack Exchange
Mathematics Stack Exchange

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 $K$.

4912 questions
1
vote
1 answer

If $F(a) \subseteq F(c)$, then $F(c)$ is an extension of $F(a)$

How do I prove the statement above? I have noticed my textbook using it but I cannot find an explanation of why that is true. I can see why this is true in things like $Q(\sqrt[4]{2})$ and $Q(\sqrt{2})$ but that is because I can find the polynomial…
Sorfosh
  • 3,266
1
vote
1 answer

How to prove that F is extension of K

I need to prove that filed F is extension of K. I know that F is isomorphic with $K[x]/(m)$, K is also some field and polynomial $m \in K[x]$ which is irreducible over K. I think, if F and K is isomorphic they have to be same. So F could be a…
aladar90
  • 23
1
vote
0 answers

Properties of intermediate fields in an extension

Let $L$ and $M$ be intermediate fields in the extension $K \subset F$. (a) $[LM : K]$ is finite if and only if $[L : K]$ and $[M : K]$ are finite. (b) If $[LM : K]$ is finite, then $[L : K]$ and $[M : K]$ divide $[LM : K]$ and $[ LM : K] \le [L :…
Tim
  • 67
1
vote
1 answer

Relationship between different splitting fields

Suppose I have a field extension $F\subset E$, with $E$ algebraically closed. For $f(x),g(x)\in F[x]$, consider $K_{1}$ the splitting field of $f(x)$ and $K_{2}$ the splitting field of $g(x)$. I'm trying to determine the condition under which…
roo
  • 5,598
1
vote
0 answers

If a root $ \xi $ of the polynomial f(x) is adjoined to $ \mathbb{Z}_{5} $

(a) Show that the polynomial $ f(x)=x^{3}+x+1 $ is irreducible over $ \mathbb{Z}_{5}[x] $ . (b) If a root $ \xi $ of the polynomial f(x) is adjoined to $ \mathbb{Z}_{5} $ , how many elements are there in the resulting field $ \mathbb{Z}_{5}(\xi)…
MAS
  • 10,638
1
vote
1 answer

Simple extension over a field $\mathbb{Q}(\sqrt{7})$

I was wondering is the field $\mathbb{Q}(\sqrt{3},\sqrt{7})$ simple extension of a field $\mathbb{Q}(\sqrt{7})$? How can we show that? Showing that the field $\mathbb{Q}(\sqrt{3},\sqrt{7})$ is simple extension of a field $\mathbb{Q}$ is easy and I…
rubin
  • 11
1
vote
0 answers

perfect squares in $\mathbb{Q}(\sqrt{2},\sqrt{91})$

Given a polynomial $P(X)$ with coefficients in $\mathbb{Q}(\sqrt{2},\sqrt{91})$ how do I find values of $X$ in $\mathbb{Q}(\sqrt{2},\sqrt{91})$ such that $P(X)$ is a perfect square in $\mathbb{Q}(\sqrt{2},\sqrt{91})$? Or how do I decide whether no…
user84909
  • 412
1
vote
1 answer

Nonisomorphic field extensions in which a polynomial has a root

Let's say I have the polynomial $f(x)=x^4 +1 \in \mathbb{Z}_{3}[x]$. I know that this polynomial is reducible in $\mathbb{Z}_{3}[x]$ since it can be decomposed as follows: $$ f(x) = (x^2 +x +2)(x^2 +2x +2) $$ where both factors are irreducible…
Ninosław Brzostowiecki
  • 1,411
1
vote
1 answer

a claim about extension of field

Let A⊂B be rings,with B integral over A and B is finitely generated over A. Let P be a prime ideal of B and p be a prime ideal of A such that p＝P∩A , then B/P is an extension of A/p of finite degree. I can't think of how to proof this claim. Why is…
vicarious
  • 11
1
vote
2 answers

Is this inheritance valid for algebraic extensions?

Let $E/F$ be an algebraic extension. Let $\sigma:F\to E$ be a homomorphism in the category of fields. Can it be shown that the extension $E/\sigma(F)$ is also algebraic?
drhab
  • 151,093
1
vote
3 answers

Degree of a field extension with three elements

I am struggling with the following: Let's have the field extension $R/Q$. How can I find $[Q(\sqrt 6,\sqrt 10,\sqrt 15):Q]$? (*) So far all I have is that: $[Q(\sqrt 6):Q] = 2$, because $minpoly_{\sqrt6}(X) = X^2 - 6$ Samely $[Q(\sqrt 10):Q] =…
K.A.
  • 619
1
vote
1 answer

find the relation $\beta = f(\alpha)$ when $\beta \in Q(\alpha)$

I am interested in finding the relation $\beta = f(\alpha)$ when $\beta \in Q(\alpha)$ and their minimal polynominals are given. For example, let $\alpha$ be $exp(2\pi i/7)$ and $\beta$ be a root of…
aerile
  • 1,437
1
vote
0 answers

If $a+b$ and $ab$ are algebraic then $a,b$ are algebraic

I need to show that if the sum and the product of two complex numbers is algebraic, then each of then is algebraic. We have that the extensions $Q(a+b)/Q$ and $Q(ab)/Q$ are finite, so the extension $Q(a+b,ab)/Q$ is finite.…
user232536
  • 11
1
vote
1 answer

What is the basis for $[\mathbb{Q}(ɛ) : \mathbb{Q}]$?

Let ɛ = e2πi / 5. $[\mathbb{Q}(ɛ) : \mathbb{Q}]$ = 4 I originally thought the dimension was 2 with basis {1, ɛ}, but it is actually 4. What exactly is the basis, and why is it not {1, ɛ}?
James Norton
  • 454
1
vote
2 answers

Why is $\Bbb Q(\sqrt[3]{2},\sqrt{-3})$ the splitting field of $x^3-2$?

I just don't see it. Too my mind, it would suffer to show that $i$ or $\sqrt 3$ are inside the extension, so that you can construct a primitive third root of unity.
Greg P.
  • 631
1 2 3
4
5 6