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. Please use (galois-theory) instead for questions specifically about that topic. Use (classical-mechanics), (electromagnetism), (general gravity) or (quantum-field-theory) for physical fields.

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Splitting field of degree $p(p+1)$ contains a Galois subextension of degree $p$.

I've been studying for an algebra qualifying exam. Any help with the following result would be appreciated. Suppose $E$ is a splitting over $\mathbb{Q}$ of an irreducible polynomial $f(x)\in\mathbb{Q}...
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1answer
53 views

To “determine” all the cubic roots of $2$, with 2 being an element in $Z_7$?

Cubic root of an element in a field? Regarding to this question, I have clarified myself about the definition of a cubic root in a field. So the next question that pops up in my mind is that to find ...
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32 views

Coordinate Transformation, vectors, tensors etc.

This is a resource recommendation question. I have started to read classical field theory, although I find it easy, some mathematics in it, I haven't encountered before. These are coordinate ...
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1answer
39 views

Cubic root of an element in a field?

In my textbook, I see the mentioning of the cubic root of an element in a field with the notation $\sqrt[3]{a}$ for $a \in F$ with $F$ being a field. I am not sure if I intepreted this correctly: $\...
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1answer
73 views

Which field does $\overline{\mathbb{C}(t)}$ refer to?

I'm reading a textbook on algebraic geometry and it mentions the field $\overline{\mathbb{C}(t)}$ without defining it. It does define $\mathbb{C}(t)$ as the field of rational functions with ...
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2answers
37 views

Is there a difference between the Galois group of $K/F$ and the Galois group of $E/K$ where $K$ is an intermediate field?

Given an extension $E/F$, with intermediate fields $E/K_1/K_2/…../F$, I want to know if $Gal(K_n/F)=Gal(E/K_n)$. By Galois correspondence it seems like this should be true, there are still the same ...
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1answer
22 views

Questions on the formulation of this proof showing an extension is simple if and only if there are finitely many intermediate fields.

My lecturer gave the following proof (I've written it word for word) to show that an extension of finite degree $E/F$ is a simple extension of $F$ if and only if the number of fields $K$ with $F\...
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1answer
24 views

Question about homomorphic image of maximal ideals

I am trying to clarify a statement regarding the homomorphic image of a maximal ideal Let $\varphi: F\rightarrow F'$ be an isomorphism of fields. The map $\varphi$ induces a ring homomorphism (also ...
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4answers
89 views

Why is $K:=\{x+\sqrt{2}y : x,y\in \mathbb{Q}\}$ a field? [duplicate]

Why is $K:=\{x+\sqrt{2}y : x,y\in \mathbb{Q}\}$ a field? I know that if $K$ is a sub-field of a field (in this case it should be $\mathbb{R}$?), it is a field itself. Firstly, have to prove that if $...
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64 views

Can I think of a field as a distributive category?

My understanding is that a field $(k, +, \times)$ is a set $k$ with abelian group structures $(k, 0, +)$ and $(k \setminus \{0\}, 1, \times)$ such that multiplication distributes over addition. Can ...
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3answers
190 views

Extending Galois and Cyclotomic Fields

In this question I asked whether, given a Galois field $K$ with Galois group $H$, it was possible to find a Galois extension $L/K/\mathbb{Q}$ of a desired degree with Galois group $G$ and $H \leq G$ ...
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38 views

Find me a field with these properties. [duplicate]

So I am searching for a field $K$ with these properties: $1.$ $K$ is infinite. $2.$ $K$ is not algebraically closed. $3.$ $K$ is algebraic over a finite field. I do not think the answer will be ...
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1answer
31 views

Showing $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})=\mathbb{Q}(\sqrt{p}\sqrt[3]{q})$. [duplicate]

Let $p$ and $q$ be prime numbers, $p\neq q$. Now I have to show $L:=\mathbb{Q}(\sqrt{p},\sqrt[3]{q})=\mathbb{Q}(\sqrt{p}\sqrt[3]{q})=:K$. The direction $K\subseteq L$ is clear. For the other ...
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36 views

Is the finite field $Z_p$ an ordered one? Or can a finite field be ordered? [duplicate]

Since residue classes modulo $p$ are not numbers but sets of numbers with several operations defined on them, i don't think we can compare them at all. Yet i'm not sure. For sake of an example lets ...
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2answers
59 views

How to find the fixed element for Galois group?

This is extension of my question of the link How to find the fixed field for Galois group? (I've already solved the question (1)) Question) Duplicated Let be $K$ the finite splitting field of $f(x) ...
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1answer
67 views

Prove that the number of $\alpha\in\mathbb{F}_{27}$ such that $|A_\alpha|=26 $ equals 12.

Let $\mathbb{F}_{27}$ denote the finite field of size 27. For each $\alpha\in\mathbb{F}_{27}$ , define $ A_{\alpha}$ $=$ { $ 1, 1+\alpha , 1+\alpha+{\alpha}^2 , 1+\alpha+{\alpha}^2+{\alpha}^3 , .... $}...
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1answer
71 views

How to find the fixed field for Galois group?

Let be $K$ the finite splitting field of $f(x) (\in \Bbb Q[x])$ over the field, $\Bbb Q$(rational number set) And say $E_H$ is a fixed field of $H\subset \operatorname{Gal}(K/Q) $. Main Question) ...
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1answer
36 views

What does the notation -S mean if S is a set?

As the title says, what does the notation $-S$ mean if $S$ is a set? For instance, in the context of something like $S \cup -S = F$, where $F$ is a field and $S$ is a subset of $F$? Thanks in advance....
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73 views

Constructing Galois Extensions of Desired Degree

Suppose that $K/\mathbb{Q}$ is an abelian Galois extension with Galois group $H$. Let $m= |H|=[K \colon \mathbb{Q}]$. Given an abelian group $G$ with $H \leq G$ and $|G|=km$, is it possible to find an ...
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2answers
48 views

How can I prove $y^2-(x-z^2)^3$ is irreducible?

Suppose $F$ is an algebraically closed field. I'm trying to find the set of singular points of the algebraic set $X=V(y^2-(x-z^2)^3)$. So I may need to prove the polynomial $y^2-(x-z^2)^3 \in F[x,y,z]...
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0answers
47 views

Purely transcendental extensions do not affect the minimal polynomial.

I've been studying for qualiying exams and recently came across this problem. Let $L/M/K$ be a field extensions with $[L:M]<\infty$. Let $A$ be a subfield consisting of all elements of $L$ that ...
5
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1answer
43 views

How can we prove the existence of algebraic closures from first-order compactness?

A devilish remark in „Aluffi: Algebra Chapter 0“[1, Remark VII.2.7] claims that $^7$ Allegedly, the existence of algebraic closures is a consequence of the compactness theorem for first-odrer logic,...
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53 views

How to show that the splitting field of $x^7 - 5$ over $\mathbb{Q}$ is equal to $\mathbb{Q}(\sqrt[7]{5}, \exp(2\pi i/7))$?

I have a polynomial $x^7 – 5$ over $\mathbb{Q}$. I know the splitting field $K = \mathbb{Q}(\alpha,\alpha\omega, …, \alpha\omega^6)$, where $\alpha = \sqrt[7]{5}$ and $\omega = \exp(2\pi i/7)$. Now, ...
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2answers
63 views

A faster way to find the quadratic extensions of a field extension?

The usual method I have for finding Galois correspondence goes like this : Say we have the Splitting field of $x^4-3$, i.e. $\Bbb Q(i,\sqrt[4]{3})$. Then it's generating automorphisms are those ...
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0answers
39 views

Difficulty understanding this definition of Galois extensions

I'm having some trouble understanding the definition of a Galois group given in class. It says that a field extension $E/F$ is a Galois extension if its group of automorphisms is such that $Fix(Aut(E/...
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3answers
55 views

$\frac{\mathbb{C[x]}}{\langle x-a\rangle}$ is isomorphic to which field?

We know that every polynomial of degree one is irreducible over $\mathbb{C}$ that is $\langle x-a \rangle$ is maximal ideal in $\mathbb{C[x]}$, hence $$\frac{\mathbb{C[x]}}{\langle x-a\rangle}$$ is ...
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1answer
1k views

Can “Taking algebraic closure” be made into a functor?

I am now confused with such problem as title goes. To be exact, the problem is Does there exist a functor from $A:\mathsf{Field}\to \mathsf{Field}$ with a natural transformation from identity ...
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1answer
59 views

Why should the kernel of a ring homomorphism be an ideal?

so my question is: Why should the kernel be the ideal of the field $F$? Here is the required theory. Here $(a)$ means an ideal generated by $a$.
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35 views

Let $\mathbb F$ be a field, find a necessary and sufficient condition on $\mathbb F$ such that the only semi-polynomial maps are the polynomials.

Let $\mathbb F$ be a field. A map $f : \mathbb F^2 \rightarrow \mathbb F$ is semi-polynomial if for every fixed $y$ the map $ x \rightarrow f(x,y)$ is a polynomial and for every fixed $x$ the map $y ...
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1answer
26 views

Union of subgroups of $\mathbb{C^*}$

Let $\mathbb{ℂ^∗}$ = $\mathbb{ℂ}$\ ${0}$ denote the group of non-zero complex numbers under multiplication. Suppose $Y_n$={$z\in \mathbb{C} |z^n=1$} then which of the following is (are) subgroups ...
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1answer
20 views

For $x$ lives in the algebraic closure of $\mathbb F_q$, $x^{q-1}=1$ if and only if $x \in \mathbb F_q$?

Let $\mathbb F_q$ be a finite field of order $q$ and let $C$ be its algebraic closure. Taking $x\in C$, I wonder if $x^{q-1}=1$ if and only if $x \in \mathbb F_q$. Of course $x \in \mathbb F_q$ ...
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2answers
76 views

Fermat's little theorem as a consequence of $(x+y)^p=x^p+y^p$ in $\Bbb Z_p$?

Exercise in a book I am currently reading asks that I prove Fermat's little theorem as a consequence of the fact that in $\Bbb Z_p$ for any $x$ and $y$, $(x+y)^p=x^p+y^p$. Although this feels like it ...
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0answers
21 views

Implications between two properties involving the ring $~K[x,y]~$

I tried to solve this problem, but I'm not sure if my solution is correct. The problem is the following one: Let $~K~$ be a field, $~f(x)~$ and $~g(x)~$ two irreducible polynomials in $~K[x]~$ with ...
19
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1answer
788 views

Proof that every field is perfect?

The following must be wrong, since it shows that every field is perfect, which I gather is not so. But I can't find the error: Suppose $E/K$ is a field extension and $p\in K[x]$ is irreducible ...
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0answers
36 views

Composition of fields and completion?

Let $L/K$, $L'/K$ are finite extensions of a number field, $LL'$ be the composition field, let $p$ be a prime in $K$, $q, q'$ be primes over $p$ in $L$ and $L'$ respectively, $q''$ be the prime over $...
5
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1answer
86 views

Explicit group decomposition

Let $E/F$ be a quadratic extension of non-archimedean fields. Let $p$ its maximal ideal and $O$ its ring of integers. I am interested in the subgroup $A$ of invertible matrices of the form $$ \left( \...
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1answer
43 views

Show that $\ I=(x^n) $ for some $n$, an element of $\Bbb N\cup\{0\}$. [on hold]

Assume $k$ is a field, $k[[x]]$ is the ring of formal power series. Let $I$ be an ideal of $k[[x]]$. Show that $\ I=(x^n) $ for some n, an element of $\Bbb N\cup\{0\}$.
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1answer
41 views

Algebraic independence over subfields and algebraic closure?

Are my following conjectures formulated linguistically and mathematically correct? 1.) $K$ be a field, $O$ a superfield of $K$, and $v_1,...,v_n$ be algebraically dependent over $K$. $v_1,...,v_2$ ...
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0answers
24 views

Finite extension of the inclusion $\mathbb F_p \to \Omega$ is injective?

Let $\Omega$ be an algebraically closed field of characteristic $p$ and $\mathbb F_p \to \Omega$ be the standard inclusion map. I wonder if for any finite extension $K/{\mathbb F_p}$, the extension of ...
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1answer
48 views

sub-resultant GCD algorithm for polynomials over a field

I'm trying to implement the algorithm for factoring polynomials over number fields on page 145 of Henri Cohen's A Course on Computational Algebraic Number Theory, but I'm not sure how the application ...
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0answers
25 views

What (mathematical) field does a (physics) superfield belong?

A superfield, $\phi(x,\theta)$, is define by a Grassman-even (i.e. commuting) function of a set of commuting variables ($x^\mu$) and a set of Grassman variables ($\theta^\alpha$ where $\theta^\alpha\...
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0answers
48 views

Do automorphisms preserve convergence of power series?

Consider a power series $$f(q)=\sum_{n=0}^\infty a_nq^n.$$ Suppose that $f$ has a positive radius of convergence. For an automorphism $\sigma\in \operatorname{Aut}(\mathbb C)$ define $$f^\sigma(q)=...
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1answer
37 views

Extension Field Question [duplicate]

Suppose $E = \mathbb{Q}(\alpha_1,\alpha_2,...,\alpha_n)$ where $\alpha_i^2\in \mathbb{Q}$ for $i=1,2,...,n.$ Prove that $2^{1/3} \not\in E$. I thought I could prove it by contradiction but I have ...
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1answer
15 views

Finitely generated module over a field is always completely reducible

Let $M$ be a finitely generated $F$-module where $F$ is a field. Let $\{m_1,....m_n\}$ be the generating set for $M$. Then, given $x \in M$, $\exists r_1,r_2,...r_n \in F$ such that $r_1m_1+....+...
3
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1answer
43 views

Proving that multiplicative groups of two fields F* are isomorphic

Considering the fields that are described by: $$\mathbb{F}_i = \mathbb{Z}_2[x] /\langle\mkern 1.5mu p_i(x)\mkern1.5mu\rangle, \enspace i=1,2 $$ where $p_1(x)=x^3 + x + 1$ and $p_2(x)=x^3+x^2+1$ I ...
3
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0answers
32 views

Is my line of reasoning correct for considering this fixed field as a simple extension over $GF(7)$?

Suppose we have the field of rational functions $GF(7)(t)$ in the indeterminate $t$ over $GF(7)$. Define $σ,τ ∈ Aut(E)$ by $σ(t) = 2t$ and $τ(t) = 1 /t$. Set $G = <σ,τ>$ and $F = Fix(G)$. There ...
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1answer
21 views

Let $G$ be a finite group of automorphisms of $E$ and set $F=Fix(G)$, Then why is $E:F$ always separable?

Let $G$ be a finite group of automorphisms of $E$ and set $F=Fix(G)$, Then why is $E:F$ always separable ? I have a feeling that it has something to do with the idea that if $E:F$ is separable hence ...
1
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1answer
39 views

A quicker way to decide if this extension is Galois.

Given $\alpha=\sqrt{1+\sqrt{2}}$, a min. poly for this element over $\Bbb Q$ is $x^4-2x^2-1$, as it's monic irreducible over $\Bbb Q$ and has $\alpha $ as a root. The roots of this min. polynomial ...
0
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1answer
52 views

What theorem is being invoked in this solution to deciding if this polynomial is solvable by radicals?

I was reading these solutions online http://campus.lakeforest.edu/trevino/Spring2019/Math331/Homework7Solutions.pdf for practice for an upcoming exam I have . But in question $2$ part $(a)$ the author ...
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0answers
47 views

Is the largest order of an element of a group always the size of the group? [duplicate]

Similar to how one can find a primitive root element of $\mathbb{Z}/p\mathbb{Z}$, I was wondering if, for any group of size $N$ that can be written as $(k\backslash\{0\},\cdot)$ where $k$ is a field, ...