Questions tagged [field-theory]

Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. This tag is NOT APPROPRIATE for questions about the fields you encounter in multivariable calculus or physics. Use (vector-fields) for questions on that theme instead.

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Does $a\neq b$ imply $a\otimes c\neq b\otimes c$ (when $c\neq 0$)?

I'm trying to understand by myself Hopf-Galois theory, and for an example to verify certain properties I need a map to be a bijection. The details of my example are not important, what I just need is ...
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Prove that $A$ is a field [duplicate]

I've trying to prove some theorem about Galois-Hopf Theory. It's details are not necessary to understand my question. I want to prove the following: Let $F/K$ be a Galois field extension. Let $A$ be ...
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$k\subset K$ field extension, $S$ transcendence base, then $K$ is algebraic extension of $k(S)$, why?

Terminology from Lang, Chapter VIII. 1. $k\subset K$ field extension and $S$ transcendence base, then $K$ is algebraic over $k(S)$. Why is that so? Lang states as it is something trivial. Surely we ...
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Confusion regarding a proof in Galois theory.

I am studying Galois theory from Malik,Mordeson,Sen.There is a theorem which is as follows: $K/F$ be a finite extension and $\alpha\in K$,then there exists a polynomial $g(x)$ and a field $E\supset K$...
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Definition of Topological Field

Wikipedia's definition of a topological field is as follows Let $(\mathsf{F}, \mathcal{T}_\mathsf{F})$ be a topological space and $(\mathsf{F}, +, \times)$ be a field. We say that $(\mathsf{F}, \...
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Choosing splitting field of $x^4+x^2+1$ over $\mathbb{Q}$

I choose my title "choosing" since I found two splitting field (I know splitting fields are unique!) Here is my solution to the question: First, observe that $$x^4+x^2+1=(x^2+x+1)(x^2-x+1)$$ ...
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2 votes
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Subgroups for Galois group of cyclotomic etension

My teacher gave me this problem as a personal homework. I need to determine if $\cos(2\pi/13)$ and $\cos(2\pi/55)\in K = \mathbb{Q}[\cos(2\pi/37),\cos(2\pi/15),\cos(2\pi/11)]$. For $\cos(2\pi/13$) I ...
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Integral closure in an infinite algebraic extension

If $A$ is a principal ideal domain and $L/Q(A)$ a finite field extension, then it follows from Krull-Akizuki theorem that the integral closure of $A$ in $L$ is a Dedekind domain. Now if $L/Q(A)$ is an ...
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Topological Field - What does it mean for operations to be continuous?

I understand the notion of a field $(\mathsf{F}, +, \times)$, of a topological space $(\mathsf{F}, \mathcal{T}_\mathsf{F})$ and of a continuous function $f:\mathsf{X}\to\mathsf{Y}$ between two ...
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What is the "gauge field" on the base space in a gauge field theory?

Suppose we have a principal $G$-bundle $P\xrightarrow{\pi} M$, and we want to consider a classical gauge field theory (with a field Lagrangian) on $M$ for this bundle, in the physics sense. In the ...
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Continuous group action and $\sigma(f(ω))=f(\sigma(ω))$

Let $G$ be a topological group(For example, absolute galois group ${\rm Gal}(\overline{L}/L)$), and $G$ acts continuously on topological field $L$, via $G\times L\to L$, $(\sigma,a)\to \sigma(a)$. ...
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Proof verification: given a field $k$, any finite group can be represented as $\rm Gal(E/F)$ where $k\subset F\subset E$ and $E/F$ is Galois.

Let $k$ be an arbitrary field and $G$ be an arbitrary finite group. It seems to me that one can construct fields $F$ and $E$ such that $k\subset F\subset E$ and $\rm Gal$$(E/F)$ is isomorphic to $G$ ...
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What is the value group of $ \Bbb{F}_q((t^{1/p^n}))$ ?(Detailed caluculation)

What is the value group of $ \Bbb{F}_q((t^{1/p^n}))$ ?(Detailed caluculation) I'm not good at calculating value group, and my only tactics is to calculate the value of each element. $|t^{p^{1/n}}u|=1/...
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The degree of simple radical extensions

Let $m \le n$ be positive integers. Does there necessarily exist a field extension $K/F$ such that $[K:F] = m$ and $K = F(u)$ for some $u \in K$ satisfying $u^{n} \in F$? In other words, given $m \le ...
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Can an algebraic field be redefined and an axiom added?

Can an algebraic field be redefined to allow division by 0 and an axiom included to define x/0 as x(1/0)? I was thinking of making a field, let's call it J for now, where numbers are written as a + bi ...
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What is the perfect closure of $ \Bbb{F}_p((t))$ and its value group?

What is the perfect closure of $ \Bbb{F}_p((t))$ and its value group ? I think perfect closure is $ \Bbb{F}_p((t^{1/p^∞}))$, and I think if so, $\bigcup_{n\geqq1}(1/p^n) \Bbb{Z}$? $ \Bbb{F}_p((t^{1/p^...
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How to prove that Quaternion's algebra over isomorphic to Mat2(Z [duplicate]

My ideas: I tried to build an explicit isomorphism, but as I think it is only possible when p = 1 (mod 4), and for p = 1 (mod 4) it get it. In my second attempt, I tried to look at them as vector ...
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Maximal algebraic independent in commutative algebra over field

In field extension, maximal algebraic independent elements in a set of generators (generate by means of fraction of generators) will also be maximal alebraic independent amoung all subsets. It is then ...
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Vanishing sums of integral linear combinations of roots of unity

Let $\{ \xi^{i} \}_{i=1}^{n}$ be $n$-th roots of unity for some positive integer $n$. It is well known that if $n$ is a prime integer, there will be $n-1$ primitive $n$-th roots of unity which are ...
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Why does Galois theory most naturally take place in the context of fields?

At least as far as I can tell, historically Galois theory was a more computational tool than it appears now, and https://hsm.stackexchange.com/questions/8099/how-did-the-modern-understanding-of-galois-...
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Divisibility of unitriangular matrices over a field of characteristic 0 [duplicate]

Definition: A group $G$ is said to be divisible if for any nonzero integer $n$ and for any $g \in G$ there exists $h \in G$ such that $g = h^n$. Let $U_n(k)$ be the subgroup of unitriangular $n \times ...
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If E is a finite algebraic extension of K, prove E(x) is finite algebraic extension of K(x)

I am trying to prove the following question: Let $K$ be a field with $E$ finite algebraic extension of $K$, $\tilde{E}=E(x_1,...,x_n)$ and $\tilde{K}=K(x_1,...,x_n)$ (fields of quotients of ...
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A lemma about the Galois group of radical extensions in Rotman's book Advanced Modern Algebra

Here is the proof of Lemma A-5.19 in Chapter A-5 of Rotman's book Advanced Modern Algebra. (What he calls a 'pure extension' is commonly called 'radical extension' by most authors.) I am confused by ...
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If a sequence is generated by a $\mathbb{Q}$-polynomial passed mod $p$, can we find an appropriate polynomial over an extension of $\mathbb{F}_{p}$?

If we have a polynomial that takes integer values for integer inputs, we can take its outputs at integer inputs and pass them $\text{mod }p$. However, my understanding is that the coefficients of the ...
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Lemma A-5.19 about the Galois group of radical extensions in Rotman's book Advanced Modern Algebra

Here is the proof of Lemma A-5.19 in Chapter A-5 of Rotman's book Advanced Modern Algebra. It is about the characterization of the Galois group of pure extensions (which are mostly called radical ...
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Clarification about field extension and its degree

I know there are some posts about this, but I'm still confused regarding this specific question. It is said that the dimension of any field extension $\mathbb{Q}(w)$ is the degree of the irreducible ...
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How do we know that $F[\alpha]$ is a ring?

Note that this question is not about the proposition $\alpha$ is algebraic over $F \iff F[\alpha] = F(\alpha)$ I've got the following note in my textbook (given without proof, suggesting the statement ...
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non isomorphic algebraically closed fields

I am trying to find all algebraically closed fields(up to isomorphism). I found that the field of all algebraic numbers over $\mathbb Q$ is algebraically closed and I also know that the field of ...
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Vector Space and Expansion of Fields

I have learnt the complexification of real vector spaces. Let $V$ be a vector space over $\mathbb{R}$, and we can define a new vector space $V_{\mathbb{C}}$ which is a complex vector space and is ...
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Does this property of certain fields have a better description? [closed]

Let $\mathbb{F}$ be a field. Then, consider the subfield of $\mathbb{F}$ generated by $1$, which is to say, every element generated by products, inverses and sums of multiples of $1$. If $\mathbb{F}$ ...
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Function Fields & Ring of regular functions

From here - https://crypto.stanford.edu/pbc/notes/elliptic/funcfield.html This leads us to define the ring of regular functions of $E$ to be $K[E] = K[X,Y]/\langle f\rangle$ Its field of fractions $...
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Show that $f(t)=t^8-2t^4+9$ is the minimal polynomial of $\alpha = \sqrt{i+\sqrt 2}$ over $\mathbb{Q}$

I am really struggling to show that. I can't find a proof for $f$ to be irreducible. Eisentsien doesn't work. Revesing doesn't lead me anywhere and mod p didn't work as well, is there any criterion I ...
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1 vote
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How can I show that $\mathbb{Q}(\alpha^3)\subsetneq\mathbb{Q}(\alpha^3\sqrt{2})$?

I am struggling with this field extensions problem. Let $\alpha \in \mathbb{C}$ be a root of $x^5+7x^2-14x+14 \in \mathbb{Q}[X]$. Show whether the following statement is true or false: $$ \mathbb{Q}(\...
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Why do we require uniqueness in the universal property for a fraction field?

I'm a university student taking a course in abstract algebra. My professor recently introduced fraction fields, giving this definition: Let $R$ be an integral domain. There exists a field $F$, called ...
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If $f(x)$ is irreducible, is $f(x^k)$ irreducible?

Let $f(x)\in\mathbb{Z}[x]$ be an irreducible polynomial of degree $\ge 2$. Is it true that $f(x^k)$ is irreducible for $k\ge 2$? If not true, under what hypothesis, we can gurantee positive answer? ...
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Solvability of the quintic by radicals - missing step?

A theorem due to Galois asserts that a polynomial $f\in F[x]$ can be solved by radicals iff. the Galois group of $f$ is a solvable group. In my lecture notes as a corollary I have the following: The ...
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I want someone to explain example 1 [duplicate]

enter image description hereenter image description here I want someone to explain example 1 clearly
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Are there any interpretations of field theory which would allow for a negative degree of a field extension?

I've only ever seen finite field extensions indicated as $[L:K] < \infty$. I've never seen $-\infty<[L:K]<\infty$. I take this to mean that field extensions of a negative degrees are not ...
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1 vote
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Every finite separable extension is contained in a Galois extension

I am having trouble understanding the following proof: Claim: Let $K/F$ be a finite separable field extension. Then $K$ is contained in a Galois extension $K \supset L \supset F$. Proof: Since the ...
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Let $\alpha = \sqrt{5 + \sqrt{5}}$. Prove that $\mathbb{Q}(\alpha)/\mathbb{Q}$ is Galois and find the Galois group [duplicate]

So far, what I have figured out is that $\alpha$ is a root of the polynomial $f(x) = x^4 - 10x^2 + 20 \in \mathbb{Q}[x]$, and that $f$ has 4 distinct roots: $$ \alpha = \alpha_1 = \sqrt{5 + \sqrt{5}},...
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Compute the Galois group for $f(x) = (x^2-p)(x^2-q)(x^2-pq)$ over $\mathbb{Q}$ and determine all subfields of splitting field

For $f(x) = (x^2-p)(x^2-q)(x^2-pq) \in \mathbb{Q}[x]$ where $p\neq q$ are primes I need to compute the Galois group for $f$ over $\mathbb{Q}$ and determine all subfields of the splitting field. Here ...
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Example of a field of characteristic p in which the Frobenius endomorphism fails to be surjective

Let $F$ be a field of characteristic $p$. Then the $p$-th power map, $x\mapsto x^p$ is called the Frobenius endomorphism. If $F$ is a finite field, this is an automorphism. Injectivity of this map is ...
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Prove that $D$ is a field. [duplicate]

Here is the question I want to answer: Let $F$ be a field and $D$ an integral domain which is a finite dimensional vector space over $F.$ Prove that $D$ is also a field. Here are my thoughts: Since $F$...
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Is $i$ an element of $\mathbb{Q}(\sqrt[4]{5},w)$ where $w=e^{2πi/3}$?

I am trying to check if $i$ is an element of $\mathbb{Q}(\sqrt[4]{5},w)$ where $w=e^{2πi/3}$. How can I check if this is the case? Would it be correct to express my field as $a\mathbb(\sqrt[4]{5}) + b ...
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$F(\sqrt d)$ is an ordered field

I'm trying to solve this exercise: (hartshorne Euclid and beyond ex 15.3) Let $F$ be an ordered field, let $d>0$, and suppose that $d$ does not have a square root in $F$. Let $F(\sqrt d)$ denote ...
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Infinitely large Galois extensions of $\mathbb Q$ inside $\mathbb Q_p$

Let $p$ be a fixed prime number and denote $\mathbb Q_p$ the field of $p$-adic numbers. For each positive integer $n$, I would like to construct a finite Galois extension $K/\mathbb Q$ of degree at ...
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Are there absolute value of field, which is not discrete in $\Bbb{R}_{>0}^×$, and also not dense in $\Bbb{R}_{>0}^×$?

Let $K$ be a field with $\Bbb{Q}_p⊆K⊆\Bbb{C}_p$. Are there absolute value of field, which is not discrete in $\Bbb{R}_{>0}^×$, and also not dense in $\Bbb{R}_{>0}^×$? All values I know is dense or ...
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If $L=K(\alpha_1,\ldots,\alpha_s)$, then each $\alpha_i$ is algebraic

I do not understand the following remark from the lecture: Let $L/K$ be a finite field extension. If $L=K(\alpha_1,\ldots,\alpha_s)$, then each $\alpha_i$ is algebraic. Could you please explain this ...
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The Galois group of an irreducible quartic whose roots are pairwise rationally independent

Let $f \in \mathbb{Q}[x]$ be an irreducible quartic, $L/\mathbb{Q}$ its splitting field. Label its roots $\alpha_1, \dots , \alpha_4$. Suppose that $$\mathbb{Q}(\alpha_i) \, \cap \, \mathbb{Q}(\...
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Two seemingly different totally ramified extension,$ \Bbb{Q}_p(ζ_{p^n})$ and $\Bbb{Q}_p(p^{1/n})$

$ \Bbb{Q}_p(ζ_{p^n})$ and $\Bbb{Q}_p(p^{1/n})$ are both totally ramified extension over $ \Bbb{Q}_p$ each has extension degree $p^n-p^{n-1}$ and $n$. The former can be regarded as Lubin Tate extension,...
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