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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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What is the probability that a rational prime remains prime in $\textbf Z[i,\sqrt{-3}]$?

Using Chebotarev's density theorem, asymptotically, what is the probability that a rational prime remains prime in $\textbf Z[i, \sqrt{-3}]$?
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1answer
17 views

Simple field extension where primitive element is algebraic

I know that when $E$ is a field extension of field $F,$ $t \in E$\ $F,$ $t$ algebraic over $F,$ than the smallest field, containing $F \cup t$ is the same as set of polynomials, with coefficients in $...
2
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3answers
34 views

Can we have two different polynomials of the same degree $d$ here in the factorisation of $x^{p^n} -x$?

In the proposition "The polynomial $x^{p^n} -x$ is precisely the product of all the distinct irreducible polynomials in $\Bbb F_p[x]$ of degree $d$ where $d$ runs through all divisors of $n$." Can we ...
1
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1answer
23 views

A question about quadratic non-residues in finite fields.

Let $p$ be a prime such that $p \equiv 1$ (mod $3$) and $p \equiv 3$ (mod $4$). Consider a quadratic non-residue $b \in \mathbb{F}_{p^2}$. Is $\mathrm{Norm}_{\mathbb{F}_{p^2}/ \mathbb{F}_{p}}(b)$ a ...
0
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1answer
13 views

How to show that this simple extension avoids adding more elements in $L$?

$F\subset L \subset K$ is a field extension, $α \in K$ is algebraic element over $F$ , whose minimal polynomial $p(x)$ over $F$ is irreducible over $L$, show that $F(α)\cap L = F$ I don't know how to ...
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0answers
19 views

Algebraic Field Extensions and Irreducible Polynomials

Let $E/K$ be a field extension, $a,b\in E$ algebraic over $K$. Show: $\text{Min}(a,K,X)$ irreducible over $K(b)$ if and only if $\text{Min}(b,K,X)$ irreducible over $K(a)$ My attemp: (1) Let deg $\...
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1answer
30 views

Unramified extension of $L(\sqrt{\alpha})/L$

I am studying an article of Chaoli and I try to understand the following statement: If $L$ is a number field and $\alpha \in L^{\times}/(L^{\times})^{2}$ then, for an odd prime $p$, $L_{p}(\sqrt{\...
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0answers
27 views

Algebraic Closure - equivalent definitions [closed]

I want to show that the following characterizations of the algebraic closure are equivalent. Let K be a field (i) $E=K$ for every algebraic extension $E/K$ (ii) Every non-constant Polynomial in $K[...
2
votes
1answer
21 views

Example of an infinite simple extension.

I'm new to Field Theory and I'm looking for an example of an infinite simple extension. Theorem: The element $\alpha$ is algebraic over $F$ if and only if the simple extension $F(\alpha)/F$ is ...
2
votes
2answers
74 views

Why must an automorphism of an extension of $\mathbb{Q}$ send 1 to a rational number?

This is from a video I was watching that claimed this: If $\phi$ is an automorphism of an extension field $F$ of $\mathbb{Q}$, then $\phi(q)=q$ for all $q\in\mathbb{Q}$. The proof started by ...
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1answer
37 views

Finding the fields between $\mathbb{Q}(\sqrt{3+\sqrt{7}},\sqrt{3-\sqrt{7}})$ and $\mathbb{Q}$

I would like to find the fields between $\mathbb{Q}(\sqrt{3+\sqrt{7}},\sqrt{3-\sqrt{7}})$ and $\mathbb{Q}$. I know the degree of $\mathbb{Q}=(\sqrt{3+\sqrt{7}},\sqrt{3-\sqrt{7}})/\mathbb{Q}$ is $8$ ...
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1answer
19 views

Degree of splitting fields

I'm learning about splitting fields but I'm not sure if I am right. Hopefully I can get some insights on whether I have been learning correctly. The question asks to find the degree of the splitting ...
1
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2answers
50 views

Describe $\Bbb{R}[x]/(x^2 + 1)$

First by previous knowledge, I do know that $\Bbb{R}[x]/(x^2 + 1) \cong \Bbb{C}$, so this might seem trivial. But I am not here to ask about that and I don't want to use this fact I am reading Dummit ...
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1answer
31 views

Finding intermediate fields of the extension $\mathbb{Q}=(\sqrt{3+\sqrt{7}},\sqrt{3-\sqrt{7}})/\mathbb{Q}(\sqrt{14})$

I want to find the intermediate of the extension $\mathbb{Q}=(\sqrt{3+\sqrt{7}},\sqrt{3-\sqrt{7}})/\mathbb{Q}(\sqrt{14})$. So far, I've been able to show that there are exactly two of them since the ...
1
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1answer
40 views

Finding the degree of $\mathbb{Q}(\sqrt{3+\sqrt{7}},\sqrt{3-\sqrt{7}})/\mathbb{Q}(\sqrt{3+\sqrt{7}})$

I would like to find the degree of the field extension $\mathbb{Q}(\sqrt{3+\sqrt{7}},\sqrt{3-\sqrt{7}})/\mathbb{Q}(\sqrt{3+\sqrt{7}})$. Here's my thoughts on this problem. I suspect the result is 2. ...
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0answers
35 views
+50

Are radical extensions splitting fields?-Confusion (need help)

Wrapping up Charles C. Pinter's "Abstract Algebra", having been introduced to the very basics of Galois theory in the previous chapter (fundamental theorem and a few other results), I'm finding myself ...
4
votes
2answers
105 views

Turning $\mathbb R^n$ into field

I am reading Apostol's fascinating text Mathematical Analysis. In a footnote on P117, he writes: If it were possible to define multiplication in $\mathbb R^3$ so as to make $\mathbb R^3$ a field ...
2
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3answers
82 views

Fundamental Question on how to prove $a \not\in K(b)$ where $a,b$ algebraic over $K$

I have a very fundamental question on how to prove something like $\sqrt{3} \not\in \mathbb{Q}(\sqrt{2})$. In all of the proofs trying to show something similar eg. here, or here it is shown that (for ...
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1answer
60 views

Major misunderstanding about field extensions and transcendence degree

So presumably this question is very basic, but I'm having some trouble with apparent contradictions in my reasoning. Let $k$ be a field and $k \subseteq K$ a field extension. We say that $K$ is a ...
2
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0answers
19 views

Left vs right degree of skew-field extensions [migrated]

Artin in his book, Geometric Algebra, says the connection between the left degree and right degree of a skew-field extension is unknown. Since I'm not an expert, I was wondering if someone knew the ...
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1answer
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Algebraically independent numbers and Archimedean field

What is the cardinality of the set of all algebraically independent numbers in $\mathbb{R}$? Can this be related to the total number of Archimedean fields possible as rational extensions of sets of ...
4
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2answers
57 views

What does it mean that field $\mathbb{F}_{p^n}$ “contains” the prime field $\mathbb{Z}_p$?

I have read in few books (example Computational Number Theory, page 77) that any extension field $\mathbb{F}_{p^n}$ "contains" as a subfield the prime field $\mathbb{Z}_p$? What exactly does "...
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1answer
25 views

A finite normal extension is also a splitting field

Let $E/K $ be a finite normal extension, then there exists $p(x)\in K[x] $ s.t. $E$ is the splitting field of $p(x) $. The proof goes as follows: $E=K(a_1,...,a_n) $ for some $a_1,...,a_n\in \Omega $ ...
2
votes
1answer
26 views

Example of a field of functions containing ln(x)

Up until now, I've mainly worked with the polynomial ring $\mathbb{R}[x_1,...,x_n]$ or the corresponding field of fractions. But I can't think of an example of a field of functions that contain non-...
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0answers
31 views

Are all finite extensions of perfect fields cyclic?

I am not well trained in number theory. If there is any mistake, please straightly edit this question or devote me if the question is too trivial. According to https://en.wikipedia.org/wiki/...
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2answers
53 views

Is it 'rare' that $a$ and $a+1$ are conjugate (= have the same minimal polynomial)?

Let $a \in \bar{k}-k$, $k$ is a field of characteristic zero and $\bar{k}$ is an algebraic closure of $k$. Denote the minimal polynomial of $a$ by $m_a=m_a(t) \in k[t]$. Is it 'rare' that $m_a=m_{...
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0answers
32 views

If $a \in L-k $ satisfies $k(a^n)=L$ (for all $n \geq 1$), then $L/k$ is Galois?

Let $k \subsetneq L$ be a finite separable field extension, and let $a \in L-k$ satisfy: For every $n \geq 1$, $k(a^n)=L$. In other words, all the non-zero powers of the primitive element $a$ are ...
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0answers
52 views

$a,b \in \bar{k}$, such that $k(a)=k(b)=k(ab)$

Let $k$ be a field of characteristic zero, and let $a, b \in \bar{k}$ ($\bar{k}$ is an algebraic closure of $k$) be two distinct elements, such that $k(a)=k(b)$. Notice that $k(a)=k(b)$ implies that ...
1
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1answer
31 views

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[...
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2answers
23 views

Motivation for the method of adjoining roots of polynomials

In Galois theory we learned the standard method of adjoining a root of an irreducible polynomial. More precisely, we saw that if $K$ is a field and $f\in K[x]$ is irreducible then the field $K[x]/(f)$ ...
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0answers
21 views

Normal extension of fields

I have to prove that $\mathbb{Q}(\sqrt{2},\sqrt{3}, u$) is a normal extension over $\mathbb{Q}$, where $u^2=(9-5\sqrt{3})(2-\sqrt{2})$. Could someone help me to prove this? I claimed that $\mathbb{Q}(\...
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2answers
50 views

Degree of extension of fixed field by infinite set of automorphisms.

If $G$ is a finite group of automorphism $E \rightarrow E$, then Dedekind-Artin theorem tells us that $[E:E^G]=\; \mid G \mid$ where $E^G$ is the subfield of $E$ fixed by the automorphisms of $G$. Is ...
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0answers
44 views

About a problem of field extension in an algebra book by Fumiyuki Terada.

I am reading an algebra book by Fumiyuki Terada. There is the following problem in this book: $E_1, E_2, K$ are fields. $K$ is a subfield of $E_1$. $K$ is a subfield of $E_2$. $p, q$ are ...
3
votes
1answer
67 views

Show a polynomial is irreducible over field $F(\alpha)$

In the field $F(u)$, let $\alpha = \frac{u^3}{(u+1)}$. Consider the subfield $F(\alpha)$ of $F(u)$. Prove that the polynomial $f(x)=x^3-\alpha x-\alpha$ is irreducible over the field $F(\alpha)$. ...
0
votes
1answer
21 views

Equality field extensions

How would you prove this: If E is an extension of K and a and b are elements of E\K, a^m belongs to K, b^n belongs to K, gcd (m, n) =1, then K(ab) =K(a,b) ?? I don't know which strategy to use to ...
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1answer
29 views

Extension with algebraic element is finite

I don't get how to proof this theorem: Let $E/K$ be a field extension. If $\alpha$ is algebraic over K, then $K(\alpha ):K<\infty$. I know that we can assume that there exists a nontrivial ...
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0answers
18 views

Finite Integral Extension of DVRs $rk_R(A)$

Let $R \subset A$ be a finite integral extension of discrete valuation rings such that $A = \oplus_{i=1} ^n R$ therefore $rk_R(A)=n$. Denote by $K_R,K_A$ the quotient fields of $R$ and $A$. Obviously ...
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1answer
60 views

Understanding of a proof

In (1) and (3), $ M' \subseteq L'$ and $H \subseteq H''$ are proved respectively. But I didn't understand how these imply $ M' \leq L'$ and $H \leq H''$ again respectively.
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1answer
34 views

Extension Degree of Fields Composite

Let $K/F$ be a field extension, $L/F$ and $M/F$ finite subextensions of $K/F$ and $LM$ the composite of $L$ and $M$. I'm trying to prove that $[LM:F] = [L:F][M:F]$ implies the trivial intersection $L\...
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1answer
37 views

transcendence basis of field extensions of $\mathbb{Q}$

In some exercice, I see the following: Let $K = \mathbb{Q}(X_1 ,\dots , X_n )$ and $k = \mathbb{Q}(e_1 , \dots, e_n )$, where $(e_i)$ are the elementary symmetric polynomials. It states: "Since K ...
4
votes
1answer
38 views

Rupture field and splitting field

Is there a characterization of irreducible polynomials over $\mathbb Q$ whose splitting field over $\mathbb Q$ are isomorphic to a rupture field? In other words, of polynomials $P \in \mathbb Q(X)$ ...
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2answers
44 views

Field Extension Equality

I want to show the following: Let $E/K$ be a field extension and $1+1 \neq 0$. Let $\alpha , \beta \in K^* $. Show $K(\sqrt{\alpha}$)= $K(\sqrt{\beta}$)$\Leftrightarrow$ $\exists$ a $\gamma \in K^*$ ...
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2answers
43 views

$f(x)$ is irreducible polynomial on field $k$, $f(\alpha)=0$. $k'$ is field extension of $k$, then $k(\alpha)\otimes_kk'\cong k'[x]/(f(x))$

$k$ is a field, $f(x)$ is irreducible polynomial on $k$, $\alpha$ is a root of $f(x)$. If $k'$ is field extension of $k$, then $k(\alpha)\otimes_kk'\cong k'[x]/(f(x))$. My idea: Since $f(\alpha)=0$,...
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0answers
61 views

Do “$K/k$ twisted” representations exist?

Given $k$-representations $V,W$ of a group $G$, where $k$ is a field, $K/k$ a field extension, if we have $V\otimes_k K\cong W\otimes_k K$ as $K$-representations, do we have that $V\cong W$? Being ...
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0answers
71 views

What is wrong with my conclusion (compositum of local fields)?

Let $K = \mathbb{Q}_2(\zeta_3)$ where $\zeta_3$ is a primitive third root of unity, and $F = \mathbb{Q}_2(\zeta_3,\sqrt[3]{2})$. Furthermore, let $L = \mathbb{Q}(\zeta_3,\beta)$ where $\beta$ is ...
2
votes
1answer
85 views

Computing degrees and ramification indices of some extensions of $\mathbb{Q}_2$

Let $K=\mathbb{Q}_2$ and $F = K(\zeta_3,\alpha)$ where $\zeta$ is a primitive third root of unity and $\alpha$ is a cubic root of $2$, i.e. $\alpha^3 = 2$.Let $K_1 = K(\zeta_3)$, $K_2 = K(\alpha)$ and ...
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2answers
38 views

The minimal polynomial is the determinant of $xI-L_{\alpha}$.

Let $K=F(a)$ a finite field extension of $F$. For $\alpha \in K$, let $L_{\alpha} : K \to K$ be the transformation $L_{\alpha} (x)=\alpha x$. Show that $L_{\alpha} $ is an $F$-linear transformation ...
3
votes
1answer
85 views

Proof of a Lemma for a local field extension with certain properties

The following result is from "Euler Factors determine local Weil Representations" by Tim and Vladimir Dokchitser: Lemma 1: Let $F/K$ be a cyclic extension of degree $n$ and ramification degree $e$. ...
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0answers
65 views

Galois group of $x^3+x+1$ over $\mathbb{Q}$ is isomorphic to $S_3 $

Let E be splitting field of $x^3+x+1 \in \mathbb{Q}[X]$ over $\mathbb{Q}$. Proof that Gal{E/$\mathbb{Q}$}$\cong S_3$. Specify all extension fields L with $\mathbb{Q} \subset L \subset E$ and the ...
1
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1answer
30 views

Weighted summation of irrational numbers

Consider the set $\{\sqrt{p_1},\dots,\sqrt{p_n}\}$, where none of $p_i$'s are perfect squares, and ${\rm gcd}(p_i,p_j)=1$ for every $i \neq j$. Prove that $0$ cannot be expressed as $\sum\limits_{k=1}...