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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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

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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1 vote
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Can form of elliptic curve digital signature equation be simpler?

I am curious why equations for signing/validating with ECDSA have forms they have. Is it possible to use simpler equation that have same properties. For example, this is an equation I found in 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?

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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x^7-1 over F_3 is not solvable by radicals [closed]

Showing that the polynomial $x^7-1$ over $\mathbb{F}_3$ is not solvable by radicals.
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1 answer
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Tower laws, intersection and product of fields

I am working on a problem sheet question of Galois theory where I get stuck: Assume all field extensions here are finite. Consider subfields $E$ and $F$ of $\Omega$. Let $EF$ denote the smallest ...
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1 answer
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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 answer
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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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4 answers
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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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2 answers
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An equivalence condition for $\mathcal{I(V}(I)) = \sqrt{I}$ over real fields

Question: Does $\mathcal{I}_{\mathbb{R}}\mathcal{V}_{\mathbb{R}}(I) = \sqrt{I}$ imply that $\overline{\mathcal{V}_{\mathbb{R}}(I)} = \mathcal{V}_{\mathbb{C}}(I)?$, where $I$ is an ideal of $\mathbb{R}[...
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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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0 votes
1 answer
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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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2 votes
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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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1 answer
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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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1 answer
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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 $a\mathbb(\sqrt[4]{5}) + b ...
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2 votes
2 answers
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$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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1 vote
1 answer
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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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0 votes
1 answer
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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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1 answer
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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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3 votes
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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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Why value group of $ \Bbb{Q}_p(μ_{p^∞})$ is $\bigcup_{1≦n}1/p^n \Bbb{Z}$?

Why value group of $ \Bbb{Q}_p(μ_{p^∞})$ is $\bigcup_{1≦n}1/p^n \Bbb{Z}$? I just need to calculate valuation of $ x∈\Bbb{Q}_p(μ_{p^∞})$, but I'm trouble representing $x$ in a explicit form. Thank you ...
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0 votes
1 answer
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Degree of elements in simple extension of $\mathbb{Q}$

Is the following statement true? Let $x$ be algebraic over $\mathbb{Q}$ and let degree of $x$ over $\mathbb{Q}$ be n. Then, simple extension $\mathbb{Q}(x)$ will be equal to the set: $$\{a_0 + a_1x + \...
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0 votes
1 answer
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Is it an extension of Galois?

Let $F \subset E \subset K$ be field extensions and $[E:F]=2$. If K/F is a Galois extension then E/F is Galois. This is true? Does anyone know how to prove it? I really appreciate any help.
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Two meanings of $0$ in field axioms for $\Bbb Z_5\times\Bbb Z_5$?

The additive identity in a field is unique, and we call it $0$. The field axioms say that every element except $0$ has a multiplicative inverse. But consider the field $\Bbb Z_5$. It has a $0$ ...
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0 votes
1 answer
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value group of $E=\Bbb{Q}_p(p^{1/e})$

I want to find what is a value group of $E=\Bbb{Q}_p(p^{1/e})$($e$ is positive integer, and this is totally ramified extension of degree $p$). I know value group of $K= \Bbb{Q}_p$ is {$p^a$|$a∈\Bbb{Z}$...
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1 answer
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What is the value group of $\overline {\Bbb{Q}_p}$ and $ \Bbb{C}_p$ ? And are they discrete?

What is the value group of $\overline {\Bbb{Q}_p}$ and $ \Bbb{C}_p$ ? And are they discrete? For finite extension of $ \Bbb{Q}_p$, there are known results for extension of valuations, but what about ...
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  • 574
0 votes
1 answer
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Finding subfields of an Extension Field [duplicate]

Let a $\in$ C be a root of the polynomial $X^4+ 1 \in Q[X]$ Consider the field extension Q(a) of Q. Find three fields $K_1, K_2,K_3$ such that $Q \subset K_i \subset Q(a)$ for i=1,2,3. I found out 2 ...
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0 votes
1 answer
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$(O_K/pO_K)^p=O_K/pO_K$holds, then $∃b∈K^×$, such that $|a-b^p|≦|p|$

Let $L$ be finite extension of $ \Bbb{Q}_p$ and field $K$ satisfies $L⊆K⊆\Bbb{C}_p$.Let $O_K$ be ring of integers of $K$. Suppose $(O_K/pO_K)^p=O_K/pO_K$・・・① holds, then $∃b∈K^×$, such that $|a-b^p|≦|...
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  • 574
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Galois group of $\mathbb{Q}(\sqrt{a+d\sqrt{b}},\sqrt{a-d\sqrt{b}})$.

I was reading these lecture notes of Miles Reid: https://homepages.warwick.ac.uk/~masda/MA3D5/Galois.pdf on page 47, he writes example 3.21 of $\mathbb{Q}(\sqrt{a+\sqrt{b}},\sqrt{a-\sqrt{b}})$, but he ...
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1 vote
0 answers
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Does the discriminant of a polynomial depend on the underlying field?

Say I want to show the discriminant of $X^4 + rX + s \in \mathbb{Q}[X]$ is $-27r^4 + 256s^3$. I can do this by showing it's a symmetric polynomial in r,s so by total degree in the roots it's a linear ...
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0 votes
0 answers
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Resolvents of quartic polynomials

Let $f(x)\in\mathbb{Q}[x]$ be monic irreducible of degree $4$, with $$ f(x)=(x-\alpha_1)(x-\alpha_2)(x-\alpha_3)(x-\alpha_4) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (\alpha_i\in\mathbb{C}) $$ In ...
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2 votes
1 answer
35 views

Compositum of intersection fields

Suppose that $K_1$, $K_2$ and $K_3$ are finite Galois extensions of a field $k$. Let $(K_1 \cap K_3)(K_2 \cap K_3)$ be the compositum of $(K_1 \cap K_3)$ and $(K_2 \cap K_3)$. Is it always true that $$...
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1 vote
0 answers
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Radical Solvability by $p$th roots

The characterization of field extension radical solvability in terms of the Galois group is fairly straightforward and well-known. I'm curious about a more specific problem. Namely, given a field $F$ ...
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2 votes
1 answer
23 views

Degree of subfield fixed by single automorphism

Let $L/K$ be a finite Galois extension. If $\sigma \in \mathrm{Gal}(L/K)$ has order $d$, is it the case that $$L^\sigma := \{ \ell \in L : \sigma(\ell) = \ell\}$$ satisfies $[L^\sigma:K] = d$? This ...
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1 vote
0 answers
49 views

Distinct characters are independent.

I am studying Galois theory.I am following the lecture series of K.Hanumanthu on Introduction to Galois Theory.He has started the topic with the definition of group characters. Defn. Let $G$ be a ...
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