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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.

13
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234 views

The possible degrees of $\mathbb{Q}(a,b)$ in terms of the degrees of $a$ and $b$

Let $n,m \geq 1$ be natural numbers. Is there a characterization of those natural numbers $d$ for which there are algebraic numbers $a,b$ of degrees $n,m$ such that $\mathbb{Q}(a,b)$ has degree $d$ ...
12
votes
0answers
306 views

What is the degree of a real closure of an ordered field?

Given a subfield $F$ of $\Bbb R$, let the real-closure of $F$ be the smallest subfield of $\Bbb R$ which is real-closed and extends $F$. Since the theory of real-closed fields is complete, this is in ...
11
votes
0answers
89 views

Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ , $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$, what is the intersection?

Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ and $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$. What is the intersection $F_\infty\cap K_\infty$? (Here $\zeta_{2^n}$ is a ...
10
votes
0answers
606 views

On a Proof in Galois Theory

We have the following lemma (used in the proof of Abel's Theorem): If $\text{Char}(F)=0$, $E/F$ is a radical extension, and $K/E$ is the Galois closure of $E/F$, then $K/F$ is also a radical ...
9
votes
0answers
240 views

Why is the degree of a rational map of projective curves equal to the degree of the homogeneous polynomials?

Let $C_1 \subseteq \mathbb{P}^m$ and $C_2 \subseteq \mathbb{P}^n$ be projective curves, and let $\phi : C_1 \rightarrow C_2$ be a nonconstant rational map given by $\phi = \left[ f_1, \ldots, f_n \...
8
votes
0answers
113 views

$m = [K(\alpha):K], n=[K(\beta):K]$ and $\gcd(m,n)=1$, prove that $K(\alpha + \beta) = K(\alpha,\beta)$

I am looking for an elementary demonstration of this: Suppose $K$ a field, $\mathrm{char}(K)=0$ and $K(\alpha) \supseteq K$ and $K(\beta) \supseteq K$ field extensions. Denote $n=[K(\alpha):K]$ and ...
8
votes
0answers
91 views

Sufficient conditions for $\mathbb{Q} (\alpha , \beta )=\mathbb{Q} (\alpha +c\beta ) \forall c\in \mathbb{Q}^*$

What are some sufficient conditions for $\mathbb{Q} (\alpha , \beta )=\mathbb{Q} (\alpha +c\beta ) \forall c\in \mathbb{Q}^*$ with $\alpha , \beta $ algebraic over $\mathbb{Q}$? We know that, for ...
8
votes
0answers
103 views

Field having exactly two extensions of each degree

It is well-known that a finite field has a unique extension of degree $n$, in a given algebraic closure, for every $n \geq 1$. Is there a field $F$ such that, in some given algebraic closure of $F$,...
8
votes
0answers
160 views

Let $\mathbb{K} $ be a field of characteristic $p>0$ and $\mathbb{F} | \mathbb{K} $ a finite and separable extension.

Let $\mathbb{K}$ be a field of characteristic $p>0$ and $\mathbb{F}/ \mathbb{K}$ a finite and separable extension. Show that if $B=\{\alpha_1,\dots,\alpha_n\}$ is a basis, then $C=\{\alpha_1^p,\...
7
votes
0answers
196 views

Galois' theory: fixed subfield formula.

In a homework dealing with Galois' theory, I am asked to prove the following standard statement, known as the fixed subfield formula: Theorem. Let $L$ be a field and $G$ be a finite subgroup of $\...
6
votes
0answers
57 views

What is the Galois group of the Hyperreal Numbers?

The Galois Group of $\mathbb{R}$ as an extension of $\mathbb{Q}$ is trivial. That’s because any field automorphism of $\mathbb{R}$ is order preserving, so since $\mathbb{Q}$ is dense in $\mathbb{R}$, ...
6
votes
0answers
184 views

Isomorphism between tensor products (base extension)

Let $A,B$ be two $\Bbb Q$-algebras. Assume that $A \otimes_{\Bbb Q} \Bbb C \cong B \otimes_{\Bbb Q} \Bbb C$ as $\Bbb C$-algebras. Does it follows that $A \otimes_{\Bbb Q} \overline{\Bbb Q} \cong B \...
6
votes
0answers
217 views

Kodaira dimension of $X \subset \mathbb{P}^n$ hypersurface of degree $d$

Let us fix some notations: $X = \{ F = 0 \} \subset \mathbb{P}^n$ is a non singular hypersurface of degree $d$ $K_X$ is the canonical bundle of $X$ $Q(X, K_X) = \{f/g \; \text{such that} \; f,g \in \...
5
votes
0answers
57 views

Visualization of quadratic rings $\mathbb{Z}[\sqrt{d}]$

The extensions $\mathbb{Z}[\sqrt{d}]$ of $\mathbb{Z}$ by the root $\sqrt{d}$ of the quadratic polynomial $X^2 - d$, $d \in \mathbb{Z}$ square-free, have degree $2$ and all have the same additive ...
5
votes
0answers
84 views

Counting elements with prescribed trace and norm in finite field extension.

Suppose $F$ is a finite field with $q$ elements, $l$ is a prime and $\tilde{F}$ is the degree $l$ extension of $F$. Given elements $a,b \in F$ with $b \neq 0$, how many elements in $\tilde{F}$ have $a$...
5
votes
0answers
124 views

In constructive mathematics, can the algebraic numbers be constructed directly from the rationals?

Is it possible in constructive mathematics (no excluded middle or Axiom of Choice) to construct the algebraic numbers directly from the rationals (in say HoTT), without first constructing a real line ...
5
votes
0answers
81 views

Calculating discriminant of elements of $\mathbb{Q}(a)$

Could someone please guide me through this one? Let $a$ be a root of the irreducible polynomial $x^3+rx+s \in \mathbb{Z}[x]$. Calculate: $disc(1,a,a^2)$. Thanks in advance
5
votes
0answers
82 views

Is it possible to have $F \not \cong F(X) \cong F(X,Y)$?

Is there a field $F$ such that $F \not \cong F(X)$ as fields but $F(X) \cong F(X,Y)$ ? An example might not be obvious to find, since it would give an exemple of non-isomorphic fields $K \not\cong L$ ...
5
votes
0answers
99 views

How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$?

$\mathbb{F}/\mathbb{Q}$ is a finite extension. Denote $\dim_{\mathbb{Q}}\mathbb{F}=n$. How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$? Thoughts: if $A=\{v_1,...,v_n\}$ ...
5
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0answers
309 views

finite field extensions: how to compute norm and trace

I'm studying abstract algebra and I'm stuck in the topic of fields. I don't understand what the following definition Let $R$ be a commutative ring and let $S$ be a commutative $R$-algebra, which is ...
5
votes
0answers
1k views

Galois group acts transitively

The question I am dealing with is: Let F be a field, $f(x)\in F[x]$ be irreducible and let $N/F$ be normal field extension. Let $$f(x) = g_1(x) \cdot \dots \cdot g_r (x)$$ be the factorization of $...
4
votes
0answers
96 views

Example of non-simple extension $K/F$ such that the purely inseparable closure of $F$ in $K$ is simple

The standard example of a finite extension that is not simple is to take $k$ to be a field of characteristic $p > 0$ and consider $k(x,y)$ over $k(x^p,y^p)$. In this case, the extension is purely ...
4
votes
0answers
53 views

Splitting field over field extenstion

this is my first question here, I'll apologise in advance for any kind of noob mistakes because I'm aware that I might to do them. I'm solving one assignment for my studies and I can't do anything ...
4
votes
0answers
41 views

Show that the $\mathbb{Q}$-homomorphism is well-defined.

Suppose that $a \in \mathbb{C}$ is algebraic over $\mathbb{Q}$ with $p(x) = \min(\mathbb{Q},a)$, and let $b$ be any root in $\mathbb{C}$ of $p$. Show that the map $\sigma:\mathbb{Q}(a) \to \mathbb{C}$ ...
4
votes
0answers
119 views

Galois groups of quartic over $\mathbb Q$ and $\mathbb Q(i)$

Let $f(x) = x^4 - 4x + 2$. I am asked to find its Galois groups over $\mathbb Q $ and $\mathbb Q(i)$. I believe I have done this, but have arrived at a result that I was not expecting, so I wanted ...
4
votes
0answers
86 views

Finding defining polynomial of a relative number field extension

Say I have some number field $K_1/\mathbb{Q}$ defined as: $$K_1 \cong \mathbb{Q}[x]/\langle f(x)\rangle$$ for some irreducible (monic, if this makes things easier) polynomial $f(x)$. Now, let $g(x)\in ...
4
votes
0answers
213 views

Field extensions $F \subset K \subset L$, $a \in L$ is algebraic over $K$, and $K$ is algebraic over $F$, then $a$ is algebraic over $F$.

I am trying to prove the following result from an elementary field extensions tutorial: Let $F$ be a field, $K$ a field extension of $F$ and $L$ a field extension of $K$. Assume that $K$ is ...
4
votes
0answers
290 views

Show that every algebraic field extension has a normal closure

I am trying to show that every algebraic extension of fields has a normal closure. Let $L/K$ be an algebraic extension. First suppose that the extension is finite. So $L=K(\alpha_1,...,\alpha_n)$ ...
4
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0answers
39 views

Uses for coefficients other than the trace and norm for an element belonging to a field.

The trace and norm are very useful as maps from a field extension to the base field since they are multiplicative/additive and have a lot of other nice properties. They can be defined as two of the ...
4
votes
0answers
66 views

dim. of $\left[\mathbb{Q}\left(\sqrt[n]{a},\zeta_{n}\right):\mathbb{Q}\right]$

I want to find dim. $\left[\mathbb{Q}\left(\sqrt[n]{a},\zeta_{n}\right):\mathbb{Q}\right]$ for positive rational number $a$ and $n^\text{th}$ root of unity $\zeta_{n}$ with assumption $\left[\mathbb{Q}...
4
votes
0answers
851 views

Find Galois group and all intermediate fields of the extension $L=\mathbb{Q}(\sqrt[4]{2},i)\supseteq \mathbb{Q}=K$

Find Galois group, all subgroups and the corresponding subfields of the following extension : $$L=\mathbb{Q}(\sqrt[4]{2},i)\supseteq \mathbb{Q}=K$$ 1) I found the degree of the extension $$[L:K]=8$$ ...
4
votes
0answers
222 views

Factoring irreducible polynomial over normal extension

Let $f$ be an irreducible polynomial over $F$ and $K/F$ be a normal extension. How to prove $f$ is factored by product of irreducible poly. over $K$ with same degree? I tried to do it by if $f_1, ...
4
votes
0answers
1k views

Minimal polynomial of a finite purely inseparable field extension

Given $F$, a field with characteristic $p > 0$, and a finite purely inseparable field extension $E$. Then prove that the minimal polynomial $f(x)$ of any $\alpha \in E\backslash F$ will be of the ...
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votes
0answers
329 views

Determining whether or not an extension is a splitting field

On an exam for my modern algebra class, we were asked to determine whether or not $\Bbb{Q}\left(\alpha\right)$ is the splitting field (here using the convention that splitting field means the smallest ...
4
votes
0answers
174 views

When the extensions Q(α,β) and Q(αβ) over Q are the same?

I would like to know in which conditions the extensions $\mathbb{Q}(\alpha,\beta)$ and $\mathbb{Q}(\alpha\beta)$ over $\mathbb{Q}$ are the same. Thanks in advance.
3
votes
0answers
48 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
0answers
28 views

Given a field extension E/F, $f(x)\in E[x]$, and $f(\alpha )\in F, \forall \alpha \in F$. Does $f(x)\in F[x]$?

Given a field extension E/F, $f(x)\in E[x]$, and $f(\alpha )\in F, \forall \alpha \in F$. Does $f(x)\in F[x]$ ? I have tried the thought that to lower its degree as below. Let $f(x)=\alpha_n x^n+...+...
3
votes
0answers
54 views

Profinite groups as Galois groups over $\Bbb C(X)$

$\newcommand{\Q}{\Bbb Q} \newcommand{\N}{\Bbb N} \newcommand{\R}{\Bbb R} \newcommand{\Z}{\Bbb Z} \newcommand{\C}{\Bbb C} \newcommand{\A}{\Bbb A} \newcommand{\ab}{\mathrm{ab}} \newcommand{\Gal}{\mathrm{...
3
votes
0answers
51 views

Abelian extension of $\Bbb C$

Is there a field extension $K / \Bbb C$ such that $\mathrm{Aut}(K / \Bbb C)$ is a non-trivial abelian group? Of course $K$ should be an infinite (non-algebraic) extension of $\Bbb C$, so it has a ...
3
votes
0answers
68 views

Which field has the property that there is an irreducible polynomial of any degree over it?

I have known that there is an irreducible polynomial for any degree over rational field. However,for other fields such as finite field and extension fields for the rational field,what is the ...
3
votes
0answers
77 views

Kähler Differential Of Submodules

Let be $F\subset E$ finite field extension, prove that $F\subset E$ is separable iff $\Omega_{E/F}=0$. ($\Rightarrow$) it's ok, but for ($\Leftarrow$) i've got some problems. I supposed $F\subset E$ ...
3
votes
0answers
72 views

Eigenvalues of an element in a field extension

I am studying Galois theory, and when talking about field extensions the eigenvalues of an element come up. For $L/K$ a finite field extension and $x \in L$ we consider $[ \times x] \in \operatorname{...
3
votes
0answers
32 views

Proving that $\sum_{v\in M_K, \,\,v\text{ archim.}} n_v=[K:\mathbb{Q}]$

Let $L/K$ be an extension of number fields, $M_K, M_L$ complete sets of representatives of places at $K$ and $L$, respectively. I'm familiar with the formula: $$\sum_{w\in M_L, \,\,w|v} n_w=[L:K]n_v\,...
3
votes
0answers
304 views

Determining subfields of cyclotomic field extension

I am trying to determine the number of subfields of $\mathbb Q (\zeta_{16})$ for which $[M :\mathbb Q]=2$ and their structure. (Where $M$ is a subfield and $\zeta_n$ is $n$th primitive root of unity ) ...
3
votes
0answers
98 views

Theorem 13, Chapter II of Marcus' Number Fields

I am trying to understand a proof of the following theorem: Theorem: Let $\alpha$ be an algebraic integer wich has degree $n$ over $\mathbb{Q}$. Then the field extension $\mathbb{Q}\left (\alpha\...
3
votes
0answers
67 views

Degree of splitting field extension of a polynomial over $K=\mathbb{F}(x)$

Let x be transcendent over $\mathbb{F}$, where $\mathbb{F}$ is a field with characteristic 2. I have a polynomial $f=t^3+xt+x\in K[t]:=\mathbb{F(x)}[t]$. Let L be the splitting field of $f$. I know ...
3
votes
0answers
61 views

Galois theorem in general algebraic extensions

I have proved for myself the following theorem, generalizing Galois theorem to general algebraic extensions. My question is: is it true, and is there some reference to this theorem in the literature? ...
3
votes
0answers
302 views

(Proof Verification) Prove that the sum of two algebraic numbers is algebraic.

Here is my proof for the statement that the sum of two algebraic numbers is algebraic: Let $\alpha,\beta$ be algebraic numbers. Then we know that $[\Bbb Q(\alpha):\Bbb Q]$ is finite. Similarly, $[\...
3
votes
0answers
60 views

How many roots does one add when considering the quotient of a polynomial ring?

The standard method (I think) for proving that decomposition fields of polynomials exist is by considering an irreducible polynomial $P$ (over the field $K$), and then noticing that $X + (P)$ is a ...
3
votes
0answers
278 views

order of automorphism group and index of finite field extension

EDIT-1: Since I come up with a proof, I change this previously unanswered question into a proof-verification question. Let $E / F$ be finite field extension. Define $G(E / F) = \{\phi: E \xrightarrow{...