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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.

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1answer
19 views

Proof of finite subfields for a finite field extension

I just was looking at an exercise which asks the reader to show that for $F \subset L$, if $L = F(\theta)$ for some $\theta \in L$ then there exists only finitely many subfields $K$ of $L$ containing ...
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0answers
30 views

Equivalence relation on field extensions, explanation

I am confused with lemma 25.13.3. Especially with the equivalence. At -2 lines of lemma 25.13.3 , the author writes, Given any set of extensions $\kappa \subseteq K_i$ there exists some field ...
3
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0answers
26 views

Relation between $[L \cap M : K \cap M]$ and $[L : K]$, and the Wantzel theorem

The well-known Gauß-Wantzel Theorem states that a real number $x$ can be constructed using straightedge and compass only if the minimal polynomial of $x$ (over the field $\mathbf Q$) has degree of ...
-1
votes
1answer
20 views

Finitely generated extension implies finitely generated field?

Let $K/F$ be a field extension. If $K$ is a finitely generated extension, namely $K=F[u_1,~\cdots,~u_n]$ for some $u_i\in K$. Then is $K$ also a finitely generated field(i.e. $K=\langle a_1,~\cdots,~...
0
votes
1answer
29 views

Prove every finite extension is a simple extension

can someone help me understand the proof of this theorem : Where $F(c)$ is the extension field of $F$ with $c$, Prove every finite extension of $F$ is a simple extension $F(c)$. I do not ...
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2answers
40 views

Zeros in Splitting Factor Rings

I am looking at the following proof that $x^3-x-1$ splits in an extension field of $F_{3}[x]$. Let us consider the field $$K = F_3[x]/\langle x^3-x-1\rangle$$ If $\theta$ is the image of $x$ in $K$, ...
8
votes
3answers
89 views

Showing that $\sqrt{5} \in \mathbb{Q}(\sqrt[p]{2} + \sqrt{5})$

I am attempting to show that $\sqrt{5} \in \mathbb{Q}(\sqrt[p]{2} + \sqrt{5})$, where $p > 2$ is prime. I have already shown that $[\mathbb{Q}(\sqrt[p]{2}, \sqrt{5}) : \mathbb{Q}] = 2p$. If needs ...
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0answers
39 views

Construction of Roots of Polynomials [closed]

I'm wondering if I can construct a root of the following three polynomials: $x^2-7x-13$ $x^8-16$ $x^4+x^3-12x^2+7x-1$ I think I can because the field extension of a root and Q is always divisible by ...
0
votes
1answer
23 views

Find the irreducible polynomial over $Q$(linear combination of primitive cubic roots)

Hi I'm student who just started the algebra. There are some question that bothering me. Let $\omega = e^\frac{2\pi i }{7}$ I've already known the $irr(\omega,Q) = w^6 +w^5 +w^4+w^3+w^2+w+1$ (...
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1answer
33 views

Prove that $\mathbb{Q}(\zeta_{2^{n+2}} + (\zeta_{2^{n+2}})^{-1})$ is a cyclic field extension of $\mathbb{Q}$ of degree $2^n$.

For a Galois theory course, I need to prove that $\mathbb{Q}(\zeta_{2^{n+2}} + (\zeta_{2^{n+2}})^{-1})$ is a cyclic field extension of $\mathbb{Q}$ of degree $2^n$. Constructing the minimal polynomial ...
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1answer
26 views

Intermediate field and Galois extension.

I have the following problem: Let $p(x)=(x^{12}-16)(x^2-3)$. Show that $K=\mathbb{Q}(\sqrt[3]{2},\sqrt{3},i)$ is the splitting field of $p$ over $\mathbb{Q}$, $[K:\mathbb{Q}]=12$ and show that ...
1
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1answer
60 views

Is there a general formula for $\left [ \mathbb{Q}(a,b):\mathbb{Q}(a) \right ]$?

I am trying to solve this problem which states as follows: Find the minimum polynomial for $\sqrt[3]{4}+\sqrt{-27}$ over $\mathbb{Q}(\sqrt[3]{2})$ and $\mathbb{Q}(\sqrt{-3})$. Here's what I have ...
2
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1answer
36 views

Map induced on fraction fields is finite

Let $\phi:R\rightarrow R'$ be an injective, finite map of integral domains. Is it true that induced map $\phi_1:\operatorname{Frac}R \rightarrow \operatorname{Frac}R'$ is also finite? Note: Finite ...
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2answers
74 views

Find the Galois group of $\mathbb{Q}(\sqrt{4+\sqrt{7}})/\mathbb{Q}$ [duplicate]

Let $E = \mathbb{Q}(\sqrt{4+\sqrt{7}})$. Prove that $E$ is a normal extension of $\mathbb{Q}$ and find the Galois group $Gal(E/\mathbb{Q})$. My attempt: Let $\alpha = \sqrt{4+\sqrt{7}}$. $\alpha$ is ...
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1answer
42 views

Is the extension $\mathbb{Q}(\alpha)/\mathbb{Q}$, where $\alpha = 2\pi i /3$, a splitting extension? [closed]

Is the extension $\mathbb{Q(\alpha)}:\mathbb{Q}$ where $\alpha=e^{{2\pi i}/3}$ splitting extension because it has degree 2?
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votes
1answer
25 views

Check that the corresponding extensions are normal

Exercise. we have following extensions: a) $\mathbb{Q(\sqrt(2),\sqrt(5))}:\mathbb{Q}$ b)$\mathbb{Q(\alpha)}:\mathbb{Q}$ where $\alpha=e^{{2\pi i}/3}$ c)$K:\mathbb{Q}$ where splitting field over $\...
0
votes
2answers
64 views

Find all roots of these two polynomials

There are only answers without any reasons in my textbook's questions. PLEASE HELP ME :( Find the root of these polynomials by using hints. Let the $w$ is complex root of $x^2+x+1$ 1.First ...
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votes
1answer
80 views

What's a minimal origami construction realizing a cube root?

The constructible numbers are those that can be achieved as lengths of line segments via compass and straightedge, starting with a segment of length $1$. The origami (constructible) numbers are those ...
1
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1answer
26 views

Simple extension finder for $x^p-2$

I believe that over $\mathbb{Q}[x]$, the splitting field of the polynomial $x^p-2$ is finite and separable, and so it must also be simple. However, I am unable to find, for a general $p$, what the ...
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votes
1answer
36 views

finite Galois extension

Let $E$ be a finite extension of $F$ and assume that $E/F$ is separable. Since $E/F$ is separable, there is a finite Galois extension $E'$ over $F$, containing $E$: $F \subseteq E \subseteq E'$. Show ...
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0answers
33 views

A problem from Hungerford(galois theory)

This is part of a question from Hungerford. Let $K$ be a field.Let $u=f(x)/g(x) $in $K(x)-K$.Assume that $gcd(f,g)=1.$ Then,I have to show that $ug(y)-f(y) \in K(u)[y]$is irreducible in $K(u).$To ...
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0answers
23 views

Prove that $F(x)/F(\frac{x^3}{x + 1})$ is simple algebraic extension and find its minimal polynomial

Prove that $F(x)/F(\frac{x^3}{x + 1})$ is simple algebraic extension and find its minimal polynomial. I have not seen such tasks before, can you give me a hint or right algorithm to solve it, please?
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0answers
78 views

A proof needed in minimal polynomial.

How do you prove this statement? If a, b are different prime numbers, the minimal polynomial of $\sqrt[n]{b}$ over the extension field $\mathbb{Q} (\sqrt[m]{a})$ of the rational number field $\mathbb{...
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votes
3answers
79 views

Inverse of $α∈Z_3(α)$ where $α^3+α^2+2=0$ [on hold]

Find the inverse of the element in the given field. The field is a finite extension F(α). Express your answer in the form $a_0 + a_1 α + \cdots + a_{n−1} α^{n−1}$, where $a_i ∈ F$ and $[F(α):F]=n$. $...
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votes
2answers
42 views

Inverse of $2 − 3i \text{ in } \Bbb Q(i).$

Find the inverse of the element in the given field. The field is a finite extension F(α). Express your answer in the form $a_0 +a_1α+ ···+a_{n−1}α^{n−1}$, where $a_i$ ∈ F and [F(α):F]=n. $$2 − 3i \...
3
votes
3answers
89 views

Prove that $x^2 − 2$ is irreducible over $\mathbb Q (\sqrt 3)$

Prove that $x^2 − 2$ is irreducible over $\Bbb Q(\sqrt 3)$. I was originally trying to use the fact that if $K=\Bbb Q(\sqrt 3)[x]/(x^2-2)$ and $[K:\Bbb Q(\sqrt 3)]=2$ then $x^2-2$ is irreducible over ...
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1answer
62 views

How to prove irreducibility of this polynomial?

Let $p,q$ be primes. Prove that $y^{n }-p$ is irreducible over $\mathbb{Q}(\sqrt[n ]q) $ . I have tried for some time, but still feel confused about how to prove it. Can anyone help me?
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4answers
86 views

Irreducible Polynomial Field Extensions with Root $\cos \frac{2\pi}{7}$

Show that $\theta = \frac{2k\pi}{7}$ satisfies the equation $\cos 4\theta − \cos 3\theta =0$ for each integer $k$. Hence find an irreducible polynomial over $\Bbb Q$ with $\cos \frac{2\pi}{7}$ as a ...
1
vote
1answer
46 views

Polynomial with Root $\pi+ei$

How do I find a polynomial with a root of $\pi +ei$ in the reals? I know how to do this with algebraic numbers, but not transcendental ones like e and $\pi$. Edit: I now realize that in the reals, ...
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1answer
14 views

$F\subseteq E\subseteq M$ is finite field extensions, if $M/F$ is normal then $M/E$ is normal

Let $F\subseteq E\subseteq M$ be finite field extensions. Assume if $M/F$ is normal. Prove that $M/E$ is normal. Attampt: I have proved that $M/E$ is algebric extension. Let $a\in M$. Let $f_T=\text{...
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votes
1answer
29 views

Find the Addition and Multiplication tables for GF(7)

Find the Addition and Multiplication tables for GF(7). I know in order to do this, I have to express it in terms of $Z_7[x]/(x+1)$, or some other irreducible polynomial of order 1. But I'm not sure ...
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2answers
43 views

Show that an extension $K/F$ with $|K:F| = 2$ is normal.

My attempt: It is well known that finite field extensions are algebraic. If $a \in F$, then $\min_F(a) = X-a$ trivially splits. If $a \in K \setminus F$, then $\{1,a\}$ is $F$-linearly ...
2
votes
1answer
38 views

tamely totally ramified extensions and the equation $x^e-pu=0$

Considering $\mathbb Q_p$ the p-adic rationals; Tamely totally ramified extensions are obtained by adjoining solutions of the equation $x^e-pu=0$, where $e$ is the index of ramification and $u\in \...
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1answer
35 views

Doubts regarding 'quadratic field can be obtained by adjoining square root'

This is from Artin's Algebra, Proposition 15.3.3: I think that we showed in the first para of the proof that degree two extension can be obtained by adjoining a square root. Then, why they put ...
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1answer
50 views

Any element in $K/\mathbb Q_p$ can be generated by some $\pi \in K$

following a sentence from "p-adic Numbers, p-adic Analysis and Zeta-functions" by Neal Koblitz, page 66: Let $\pi \in K$ where $K$ is an extension field of $\mathbb Q_p$ (the p-adic rationals) and $...
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3answers
40 views

Degree of splitting field of $x^3-5$ over $\mathbb{F}_7$

Find the degree of the splitting field of $f(x):=x^3-5$ over $F:=\mathbb{F}_7$. Attempt: $f$ is irreduicible in $F[x]$ (suppose in contradiction it is reducible, thus it splits to at least one ...
4
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2answers
70 views

Minimal polynomial of extension of degree 2 over a finite field with characteristic 2

I'm struggling to solve the following question. Let $F$ be a finite field with characteristic 2 and $L/F$ be a finite extension with $[L:F]=2$. Prove that there exists $\alpha\in L$ such that $L = ...
3
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0answers
48 views

Prove that $E=\Bbb Q(\sqrt[d]{a})$ [duplicate]

Let $K=\Bbb Q(\sqrt[n]{a})$ where $a \in \Bbb Q^+$ and suppose $[K:\Bbb Q]=n$(i.e., $x^n-a$ is irreducible). Let $E$ be any subfield of $K$ and let $[E:\Bbb Q]=d$. Prove that $E=\Bbb Q(\sqrt[d]{a})$ ...
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3answers
154 views

$\sigma(a)$ is a root of the minimal polynomial for $\alpha$ over $F$? [closed]

If $K/F$ is a field extension and $\alpha \in K$ is algebraic over $F$, then for any automorphism $\sigma \in \operatorname{Aut}(K/F)$, $\sigma(a)$ is a root of the minimal polynomial for $\alpha$ ...
0
votes
1answer
50 views

What does it mean for a field to be generated by some set of elements?

I've read up online, but I'm having trouble understanding what it means for a field $F$ to be generated by a set of elements $S = \{\alpha_1, ..., \alpha_n\}$ over another field $K$. What are the ...
0
votes
1answer
35 views

is $Q(π)$ simple extension of $Q$ ,where $Q$ is field of rationals? [closed]

Also suppose there is field $F$ and $α$ is not root of any polynomial in $F$ .Then is $F(α)$ simple extension of $F$?
3
votes
2answers
64 views

What is the tower of fields a number must be in in order to be constructed by MARKED RULER and compass?

I know that for unmarked ruler and compass a number (distance) is constructible from $\mathbb Q$ iff it lies in a finite tower of field extensions $\mathbb Q=K_{0}...K_{n}$, where $[K_{i} :K_{i+1}] =2$...
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1answer
45 views

Let $F = Z_5[x]/(x^3-x^2-1) = Z_5(u)$ where $u = [x]$ Give a basis for $F$ over $Z_5$

How can you have $Z_5([x])$ when x is not any particular element?
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0answers
38 views

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 ...
3
votes
1answer
40 views

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 ...
0
votes
1answer
47 views

A question on the Galois correspondance of $\sigma^2$ in $D_8$.

In the context of Galois theory the splitting field of $x^4-3$ is isomorphic to $D_8$. Therefore one of the elements of this group is $\sigma$, which maps $i\rightarrow i$ and $\sqrt[4]{3}\rightarrow ...
2
votes
2answers
38 views

Find degree of extension $Q$($\sqrt{1+\sqrt{-3}}$ + $\sqrt{1-\sqrt{-3}}$) over $Q$

I tried solving this textbook problem.Any hint how to simplify or find the degree of extension in this case ?I guess maximum degree can be 4.
2
votes
1answer
58 views

If $x$ and $y$ are complex numbers and $x+y$ , $xy$ are algebraic numbers then how to prove that $x$ and $y$ are also algebraic numbers?

I tries basic operations like multiplication and addition in a hope that i will get $x$ and $y$ out of $x+y$ and $xy$ but that didn't worked for me.Also i tried assuming a polynomial with rational ...
0
votes
0answers
5 views

If $k(x,y) = \sum_r^n(\prod_{k\not =r}^n g_k(x) )*(f_r(x)) y^r$ is nonzero in the powers of $y$, is it nonzero in the powers of x also?

Let $u,v \in F/K$, a field extension of $K$ s.t $v$ is transcendental over $K$, and assume we know that for $$q(x) = \sum_r^n (\prod_{k\not =r}^n g_k(u) )* (f_r(u) x^r \in K(u)[x],$$ $q(v) = 0$, where ...
0
votes
1answer
23 views

Prove that $[E(u): F(u)] \leq [E:F]$.

Given tower of fields $K\supseteq E\supseteq F $, prove that for $u\in K$ algebraic over $F$ and $[E:F]$ finite. This is a problem from W. Keith Nicholson's book. My idea is that $[E:F]$ is finite ...