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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2
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1answer
23 views

How to compute generators in biquadratic extension

I'm working in this setting: $K = \mathbb{Q}[\sqrt{2},\sqrt{3}]$, and let $K_1= \mathbb{Q}[\sqrt{2}],K_2= \mathbb{Q}[\sqrt{3}],K_3= \mathbb{Q}[\sqrt{6}]$ the three subfields. I know that $2 \in \...
-1
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1answer
21 views

Finite dimension implies A[a]=A(a)

I was trying to prove following statement: Let $A \subseteq F$ be field extension and $a \in F$. Then $A[a] = \{f(a)\,|\,f \in A[x]\}$. Prove that if $A[a]$ is finite dimensional as vector space over ...
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3answers
26 views

Algebraic Extensions of Z_7 with polynomials?

I am studying fields and rings and I came across this statement in a textbook that I am having trouble visualising; $\mathbb{Z_7}/\left<x^2-3\right>$ is an algebraic extension of $\mathbb{Z_7}...
2
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1answer
58 views

An extension corresponding to a subgroup of Galois group

Let $G$ be the Galois group of $f(x)=x^6-2x^4+2x^2-2$ over $\mathbb{Q}$. Describe an extension corresponding to any of it's proper subgroups of maximal order (i.e. find generators of this extension). ...
4
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2answers
60 views

How to find the degree of the extension $[\mathbb{Q}(\sqrt[4]{3+2\sqrt{5}}):\mathbb{Q}]$?

How to find the degree of extension for $[\mathbb{Q}(\sqrt[4]{3+2\sqrt{5}}):\mathbb{Q}]$? I believe that the minimal polynomial of $\sqrt[4]{3+2\sqrt{5}}$ is $x^8-6x^4-11$, but I don't know how to ...
0
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1answer
61 views

Proving that the extension field $\mathbb{Q}(\sqrt{2})$ is indeed a field

Every element in the extension $\mathbb{Q}(\sqrt{2})$ of the rational numbers can be written in the form $\alpha+\beta\sqrt{2}$ with $\alpha,\beta\in\mathbb{Q}$. The book states that this, like $\...
1
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1answer
68 views

Why is $1,a,a^2,…,a^{n-1}$ linearly independent?

I have a basic question about the proof of "Every finite field extension is algebraic". Given the extension $K\subset L$ with $n:=[L:K]$ and $a \in L$, the proof says, that we have a linearly ...
1
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1answer
42 views

Flat extension of local rings with a specified extension of residue field [closed]

Let $(R, \mathfrak m_R, k)$ be a Noetherian local ring and $K$ be a field containing $k$. Then is it true that there is a Noetherian local ring $(S, \mathfrak m_S)$ and a flat ring homomorphism $f: ...
-1
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1answer
29 views

$Q\subset L$ with $G := \text{Gal}(L/Q)$, Is $L$ contained in the field of constructible numbers? [closed]

$Q \subset L$ is a finite Galois extension with $G := \text{Gal}(L/Q)$ and $G$ is isomorphic to $S_3$, the symmetric group on $3$ elements. Is $L$ contained in the field of constructible numbers?
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1answer
39 views

Minimal extension field of $\mathbb{F}_2$ such that

Find the minimal extension field of $\mathbb{F}_2$ such that this extension contains an element of order $21$? Attempt: I know that such an extension of $\mathbb{F}_2$ is like $\mathbb{F}_{2^s}$ and $...
0
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0answers
26 views

Index of multiplicative group in field extension [duplicate]

Let $k\subset K$ a field extension. Assume that $K$ is infinite. Then the index of $k^∗$ in $K^∗$, $[K^∗:k^∗]$ (as groups) cannot be finite. How can I prove this? If $k$ is finite then I know how to ...
2
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0answers
70 views

Multiplicative group of field extension over base field

Let $k\subset K$ a field extension. Assume that $K$ is infinite. Then the index of $k^*$ in $K^*$, $[K^*:k^*]$ (as groups) cannot be finite. How can I prove this? If $k$ is finite then I know how to ...
0
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1answer
22 views

Why do we have $Q(\alpha, z) = \mathbb{Q}(\alpha, z, \overline{z})$ where $\alpha, z, \overline{z}$ are the roots of $X^3+X+1 \in \mathbb{Q}[x]$?

We are given the polynomial $f = X^3+X+1 \in \mathbb{Q}[x]$. It is easy to show that $f$ has only one real root, call it $\alpha$, and the other two roots are complex conjucates: $z, \overline{z}$. ...
0
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1answer
39 views

Find irreducible factors without factorizing [closed]

I have an exercise from my course notes that states: Find how many irreducible factors has $f(x) = x^{26}-1$ over $\mathbb{F}_3$ and their degrees. (don't factorize it) I see immediately that the $...
1
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0answers
22 views

Representatives of residue classes as elements of a vector space

Let $K$ be a number field of degree $n$. Let also $\mathcal{O}(K)$ be the ring of integers of $K$ and let $B$ be a basis for $\mathcal{O}(K)$. Given $q\in K$ we write $\mathbf{q}$ for the column ...
5
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0answers
46 views

Scalar restriction of bilinear maps

Let $R$ be a ring (commutative and with unity), $S\subset R$ be a subring. Consider three $R$-modules $M$, $N$ and $Z$. Let $\operatorname{Hom}_R(M\otimes_RN;Z)$ be the R-module of $R$-bilinear maps ...
1
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1answer
23 views

Splitting field $L$ of polynomial $f \in K[x]$ with degree $n$ satisfies $[L:K] | n!$

Suppose $f \in K[x]$ is a polynomial with degree $n$, $f = (x-\alpha_1)...(x-\alpha_n)$ over the algebraic colsure. Let $L=K(\alpha_1,...,\alpha_n)$ be the splitting field of $f$. Prove that $[L:K]$ ...
0
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0answers
28 views

Degree of an exclusionary Field Extension

Let's say I've got a field $\mathbb{Q}[i]$\ $\mathbb{Q}$. What's the degree of the field extension $\mathbb{Q}[i]$\ $\mathbb{Q}$ : $\mathbb{Q}$? Clearly without the exclusion this has a degree of 2; ...
4
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1answer
69 views

Prime decomposition of pR in $\mathbb{A}\cap \mathbb{Q}[\alpha]$ for $\alpha={^3\sqrt{hk^2}}$ if p is a prime such that $p^2|m$

I'm going through Marcus number Field chapter 3 an I'm finding very hard to understand the part about the decomposition of pR (theorem 27) that tells us that if $p\not||R/Z[\alpha ]|$ then we can ...
1
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1answer
77 views

Why this polynomial reducible? (composite field)

In galois field of prime 2, in composite field $GF((({2}^2)^2)^2)$, There are irreducible polynomials and reducible polynomials. $GF(2^2):Q_1(x) = x^2+x+1,$ $GF((2^2)^2):Q_2(x) = x^2+x+\phi,$ $\...
1
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1answer
34 views

Galois group of field extension

I was asked to find the Galois group of the extension $\mathbb{Q}(\sqrt[3]{2},\sqrt{2},e^{\frac{2\pi i}{3}})$. Since the degree of the minimal polynomials of $\sqrt[3]{2},\sqrt{2}$ and $e^{\frac{2\pi ...
0
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1answer
50 views

Finding a minimal polynomial of a root of unity over a field extension

I'm trying to find the minimal polynomial of the seventh root of unity over the field $Q(i\sqrt{7})$. I know how to do this over the rationals and have proceeded to finding that $(x-1)(x^6 +x^5 +x^4 +...
0
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1answer
50 views

Non-normal and inseparable extension of degree 5

I know that it is possible to have a non-normal and inseparable extension of degree $pq$ for any prime $p$ and any natural number $q \ne 1$. But is there possible to construct such extension of degree ...
0
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1answer
42 views

If $K(\zeta,\beta)/K$ is the splitting field for $f$, what can we say about $K(\zeta,\beta)/K(\zeta)$?

Problem setup: Let $n\in\mathbb{N}$ and let $K$ be a field whose characteristic does not divide $n$. The splitting field of $f=x^n-c$ ($c\not=0$) over $K$ is $K(\zeta,\beta)$ where $\zeta$ is a ...
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0answers
38 views

Find basis of $\mathbb Q(\sqrt[3]{2},\sqrt{-3})$ over the field $\mathbb Q(\sqrt[3]{2}\omega)$.

Splitting field for the polynomial $\ x^3-2$ over $\mathbb Q$ is $\mathbb Q(\sqrt[3]{2},\sqrt{-3})$. Now roots of the above polynomial are $\sqrt[3]{2}, \sqrt[3]{2}\omega, \sqrt[3]{2}{\omega}^2$ Since ...
0
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1answer
20 views

Show that a field extension is a radical extension

I don't really understand the notion of a radical extension, and have been struggling to figure out how to show that $\mathbb{Q} \subset K $ is a radical extension when $K = \mathbb{Q}(\sqrt[5]{1+\...
1
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1answer
24 views

Is every field a field extension of some form.

I am new to finite field theory .While I was going through the theory I figured that $\mathbb{C}$ is in fact $\mathbb{R}(i) $ isomorphic to $\mathbb{R}[x]/(x^2+1)$ .So I had a question in mind is ...
3
votes
0answers
101 views

Galois Group of $x^{6}-2x^{3}-1$

I was trying to compute the normal closure of $\mathbb{Q}[\alpha]$, where $\alpha = \sqrt[3]{1+\sqrt{2}}$. I had a reallyyyy hard time proving that $x^{6}-2x^{3}-1$ is irreducible. I proved that it ...
-2
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1answer
39 views

How to show $E\otimes_k\bar k$ has at least two prime ideals? [closed]

Suppose $k\subsetneqq E$ are two fields, and $E$ is separable over $k$, $\bar k$ is the algebraic closure of $k$, how to show $E\otimes_k\bar k$ has at least two prime ideals?
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0answers
32 views

Prove that $\mathbb{Q}[\sqrt[3]{p}, \sqrt[3]{p^2}]$ is a field.

$$\mathbb{Q}[\sqrt[3]{p}, \sqrt[3]{p^2}] := \{a+b\sqrt[3]{p}+c\sqrt[3]{p^2}: a,b,c \in \mathbb{Q}\}$$ That's the description of the wanted set of numbers. a, b, c are its elements. If the note isn't ...
0
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1answer
22 views

Calculate the degree of a composite field extension

I am working on this problem : Let $a>1$ be a square-free integer. For any prime number $p>1$, denote by $E_p$ the splitting field of $X^p-a \in Q[X]$ and for any integer $m>1$, let $E_m$ ...
-1
votes
1answer
44 views

If $K=\{a+b\sqrt2\mid a,b\in\mathbb{Q}\}$, find $[K ∶\mathbb{Q}]$ and $[K(\sqrt3) ∶\mathbb{Q}]$. [closed]

I'm new to the subject and struggling to understand the steps when finding the degree of a field extension, I've been finding the minimal polynomial and then using the degree of that as the answer, ...
0
votes
1answer
86 views

On a special kind of Discrete Valuation ring

Let $(R,\mathfrak m, \kappa) $ be a Discrete Valuation ring (i.e. a local PID , https://en.m.wikipedia.org/wiki/Discrete_valuation_ring ) containing a field $k\hookrightarrow \kappa$ such that $\...
0
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0answers
5 views

Are there upper bounds for the genus of an extension of function fields, can you have unbounded genus growth?

If $F$ is a function field with constant field $K$, and $E$ is a finite extensions of $F$, then Riemann-Hurwitz gives a way to compute the genus, $g_E$, of $E$ from the genus, $g_F$, of $F$ so long as ...
2
votes
2answers
66 views

Can there exist a finite extension $K$ where $K$ is Galois over $Q(i)$ but K is not Galois over $Q?$

Can there exist a finite extension $K$ where $K$ is Galois over $\mathbb{Q}(i)$ but $K$ is not Galois over $\mathbb{Q}$? I am trying to come up with a specific example to show it is possible. My ...
0
votes
1answer
62 views

Linearly independent vectors over a field and its subfield

Let $\mathrm{F}_q$ be a subfield of $\mathrm{F}_{q^m}$. $\mathrm{F}_{q^m}$ can be seen as an $m$-dimensional vector space over $\mathrm{F}_q$. Let $v_1,\ldots, v_k \in \mathrm{F}_{q^m}^n$ be linearly ...
0
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1answer
41 views

Valuation ring, containing a Noetherian local domain, whose fraction field is a finite extension of that of the Noetherian domain

Let $R$ be a Noetherian local domain with fraction field $K$. Let $L$ be a finite extension field of $K$. Let $V$, containing $R$, be a valuation ring with fraction field $L$. My question is: Is $V$ ...
0
votes
0answers
11 views

Problem about Splitting fields and degree of extensions [duplicate]

Here's a little problem that I'm stuck with: Prove that if $ f \in K[X] $ with $deg(f)=n$ and $ L $ is a splitting field of $ f $ over $ K $, then $[L:K]$ divides $ n! $. When I tried this problem, ...
2
votes
1answer
41 views

Calculate the degree $[K:\mathbb{Q}]$

Let $a\in \mathbb{C}$,$a^2=\frac{3+i\sqrt{7}}{2}$ and $K=\mathbb{Q}(a)$.How can we calculate the extension degree $[K:\mathbb{Q}]$ without using minimal polynomials.Its obvious that $a$ is root of $f(...
1
vote
1answer
35 views

Field extension of degree three

It's well known that $$\mathbb{Q}(\sqrt{2},\sqrt{5}) = \mathbb{Q}(\sqrt{2} + \sqrt{5})$$ This property is also true with the cubic root (for a great general theorem), but I want to prove this via an ...
-1
votes
1answer
8 views

How are these two lemmas about simple algebraic extensions and polynomials restatements of each other?

Lemma 5.14. says Let $K(\alpha) : K$ be a simple algebraic extension, let the minimal polynomial of $\alpha$ over $K$ be $m$, and let $\partial m = n$. Then $\{1,\alpha,\ldots,\alpha^{n-1}\}$ is ...
0
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0answers
53 views

The number of intermediate field extensions $\mathbb{Q}(ζ_{630})/M/\mathbb{Q}$ of $[M:Q]=3$

I'm trying to solve this exercise: Find the number of intermediate field extensions $\mathbb{Q}(ζ_{630})/M/\mathbb{Q}$ satisfying $[M:Q]=3$. Since \begin{gather} Gal(\mathbb{Q}(ζ_{630})/\mathbb{Q})...
0
votes
1answer
34 views

Degree of separability equals degree of separable closure

For a field $K$ of characteristic $p > 0$, and a finite field extension $L/K$, let $K_s$ be the separable closure of $K$ in $L$. I am to show that $[L : K]_s = [K_s : K]$. This would mean ...
1
vote
1answer
33 views

$E_1/F$ and $E_2/F$ finite field extensions, does degree of compositum $E_1E_2$ over $F$ divide the product $[E_1:F] [E_2:F]$?

Suppose $E_1/F$ and $E_2/F$ are finite field extensions. The degree of the composite field $E_1E_2$ over $F$ is less or equal to the product of the degree of $E_1$ over $F$ times the degree of $E_2$ ...
3
votes
1answer
21 views

$L/K$ field extension, $\alpha\in\overline{K}$, does $\deg f_L^\alpha \mid \deg f_K^\alpha$?

Let $L/K$ be a field extension. Let $\alpha\in\overline{K}$. We know that $f_L^\alpha \mid f_K^\alpha$, do we also have $\deg f_L^\alpha \mid \deg f_K^\alpha$? Usually $f\mid g$ does not imply $\deg ...
0
votes
0answers
29 views

If $F\subseteq K \subseteq J \subseteq L$, then $[L:F]=[F:K][K:J][J:L]$?

It is a known theorem that if $F\subseteq K \subseteq L$ are fields, then $[L:F]=[F:K][K:L]$. This may be a stupid question, but can we say the same if we have $F\subseteq K \subseteq J \subseteq L$ ...
1
vote
1answer
32 views

showing $x^4+x^2+x+1$ is reducible in $GF(81)=GF(3^4)$

I am trying to show that in $GF(81)=GF(3^4)$, $$x^4+x^2+x+1$$ is reducible I proved that it was irreducible in $\mathbb{Z}_3$ How can I prove this ? More generally, how to prove if a polynomial ...
3
votes
1answer
56 views

Proposition 14.21 in the book “Abstract Algebra” by Dummit and Foote

Page number:$592$. In the Chapter Galois Theory, I have a doubt in one step of the following proposition: Proposition 21 Let $K_1$ and $K_2$ be Galois extensions of a field $F$. (1)The intersection $...
0
votes
1answer
34 views

Galois solvable and Galois abelian elements

Let $F$ be a field of characteristic zero. Let $α$ be an element of some extension field of $F$ that is algebraic over $F$. Say that $α $ is Galois solvable over $F$ if $F(α)$ is Galois over $F$ and $...
0
votes
0answers
33 views

What is an algebra “minus” a field?

I'm trying to understand the proof of the Frobenius's Theorem: "If $A$ is an algebraic division algebra over $\mathbb{R}$ we get $A$ is isomorphic to $\mathbb{R},\mathbb{C},\mathbb{H}$" The case $\...

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