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.
13,232
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Sum of products of Fourier coefficients in finite field.
Let $\mathbb{F_q}$ be some finite field and let $f,g: \mathbb{F_q} \to \mathbb{C}$.
By $\hat{f}, \hat{g}$ let's denote the Fourier coefficients of $f,g$ with respect to the additive characters of the ...
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Why K($a_1,a_2,…,a_n$)=K[$a_1,a_2,…,a_n$] implies $a_1,a_2,…,a_n$ are algebraic on K?
I want to know how to use induction to proof K($a_1,a_2,…,a_n$)=K[$a_1,a_2,…,a_n$] implies $a_1,a_2,…,a_n$ are algebraic on K. The case n=1 is easy, but then I fail to use induction.
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Extending field isomorphism to automorphism in algebraic closure
Let $L,E$ be extensions (you may assume finite) of a field $k$, and let $\sigma: L\to E$ be a $k$-isomorphism, then there exists $\phi\in \text{Aut}(\overline k/k)$ such that $\phi|_L=\sigma$. I am ...
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Specific example for fundamental theorem of galois theory
Consider a Galois Extension $E/F$ where $F=\mathbb{Q}$ and $E=\mathbb{Q}(2^{1/3},e^{\frac{2 \pi i}{3}})$. Find all intermediate subextensions $F \leq K \leq E$ and all subgroups of $Gal(E/F)$. Then ...
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Constructing the splitting field of reducible polynomials over finite fields
*I am trying to tackle the following problem without resorting to Galois theory using explicit construction.
Constructing the splitting field of reducible polynomials over finite fields (*)
Now, to ...
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Proof of the principle of induction [duplicate]
I will be referencing the proof I provided here.
I don't understand the remark of a person, who states the incorrectness of such proof. From what I could understand, the proof is also valid not ...
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Is $1\ne0$ necessarily true in any field? [duplicate]
Given a field $(F,+,\cdot)$, and being $0\in F$ such that $a+0=0+a=a$ for each $a\in F$ and $1\in F$ such that $a \cdot 1 = 1 \cdot a = a$ for each $a\in F$, does it follow from the field axioms that $...
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Is there anywhere a list of the groups $Gal(\mathbb{Q}(\xi_n)/\mathbb{Q})$? [duplicate]
As a math undegraduate, I was doing my homework of fields and Galois theory course and observed that many questions asked us to calculate the group $Gal(\mathbb{Q}(\xi_n)/\mathbb{Q})$, where $\xi_n$ ...
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F is a field of char $0$,F(X) is the field of rational function,why for every rational function $f(X)$ ,$f(X) \neq 0$, $[F(X):F(f(X))]$ is finite?
Proposition: F is a field of char $0$,F(X) is the field of rational function,for every rational function $f(X)$,$f(X) \neq 0$ , $[F(X):F(f(X))]$ is finite.
I Know that we can prove it by letting $f(X)=...
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Polynomial over $\mathbb{Q}$ has all non real roots
Let $f\in \mathbb{Q}[x]$ such that $f$ is irreducible and $a\in \mathbb{C}\backslash\mathbb{R}$ be its root. Assume $\mathbb{Q}(a)|\mathbb{Q}$ is Galois. I want to show all roots of $f$ are non real. ...
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What does Artin mean by "real numbers are the *only* ones needed for the usual for the usual algebraic operations?"
In page 81 of the 2nd edition Michael Artin's Algebra, he introduces fields and presents $\mathbb{R}$ as a familiar example, but goes on to say that "the fact that they are the only ones needed ...
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Axioms of Field
I am currently exploring the fundamental of field theory, especially, in its connection to monoid and group.
One way we can describe a field $\mathbb{F}$ is using the following axioms:
F1. $(\mathbb{F}...
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How to calculate kink center of mass [closed]
I have a kink soliton of $\phi^4$ model:
$$ \phi(x,t)=\tanh(x-ct)$$
where $c$ is the velocity, I want to calculate the kink center of mass but I don't know how ?
any help please
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Is the class of separable extensions distinguished?
We know, thanks to embeddings, that the class of separable extensions verified the property of the fields tower. That is, in a fields tower $K\subseteq F\subseteq E$, it is true that: $E/K$ is ...
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If $f\in \mathbb{Q}[x,y]$ factors in some field extension of $\mathbb{Q}$, then it also factors in some finite extension of $\mathbb{Q}$? [closed]
If $f(x,y)\in \mathbb{Q}[x,y]$ factors non-trivially (as a product of two non-constant polynomials) in some field extension of $\mathbb{Q}$, then does it also factor non-trivially in some finite field ...
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Is there a unique irreducible polynomial relating two algebraic numbers beyond their minimal polynomials?
Given two algebraic numbers a,b over $\mathbb{Q}$, with $p(x) \in \mathbb{Z}[x]$ being the minimal polynomial of $a$, and $q(y) \in \mathbb{Z}[y]$ the minimal polynomial of $b$, suppose that there ...
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Surjective polynomial map from $\mathbb{F}_{q^2}$ to $\mathbb{F}_q$
Let $f \colon \mathbb{F}_{q^2} \to \mathbb{F}_q$ be a surjective polynomial map such that $f(x)=\prod_{i=1}^r (x - \alpha_i) \in \mathbb{F}_{q^2}[x]$ is a polynomial where $\alpha_1, \dots, \alpha_r \...
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Why the finite field $F_{p}$ being the fixed field of Frobenius map implies $\mathrm{Gal}(F / F_p)$ is a cyclic group generated by the Frobenius map
I'm reading the proof of Theorem 6.5 from the book Field and Galois Theory by Patrick Morandi
I get stuck on reading the last part of this proof. In the book, it uses $\mathcal{F}(\sigma)$ to denote ...
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If $K$ is a perfect field, then the invariant field of all its automorphism is a perfect field too.
I was asked as an exercise to proove the statement above:
If $K$ is a perfect field, then the invariant field for its automorphism group is a perfect field too.
So, if $G=Aut(K)$ and $K$ is a ...
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Question about proof of theorem regarding Galois Extensions and Intermediate Fields
In a proof in my textbook, one step of the proof reads " by the proof of Lemma 54, every injective map $υ:B\rightarrow E$ fixing $F$ (here, $F \subset B \subset E$ are all fields) must permute ...
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$\mathfrak{p}$-adic valuation of a norm
Given extension of Number fields L/K and prime ideal $\mathfrak{P} \in O_L$ lying over $\mathfrak{p} \in O_K$. Let $\hat{L}$ and $\hat{K}$ be the completions of L and K respectively. And let e and f ...
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If $\mathfrak q$ is a $P$-primary ideal, and $x \notin \mathfrak q$ then $(\mathfrak q:x)$ is $P$-primary.
I’m trying to understand the following proof of the following result. I certainly understand it, but I am confused about certain aspects of the proof.
Let $A$ a ring commutative with identity. Let $\...
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A question about Dummit & Foote's explanation on the resolvent cubic and the Galois group
I am at the beginning of my study of field theory and I am reading page $\textbf{615}$ of $\textbf{Dummit & Foote}$, and the part where I have question about is shown below:
Here for part $\...
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Algebraic field extensions are the (filtered) colimit of finite field exensions
Recently I have been looking into the module of Kaehler differentials and found that in the case of finite field extensions, $L/K$ is separable if and only if $\Omega_{L/K}^1=0$. There is also a way ...
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Irreducibility of $X^4-\sqrt{2}$ over $\mathbb{Q}(\sqrt{2})$.
To prove that $X^4-23$ is irreducible in $\mathbb{Q}[X]$ we can do the following:
We use the Eisenstein criterion with $a=23$ to see that it is irreducible in $\mathbb{Z}[X]$, and then we conclude ...
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what is the meaning of $F(S)[X]?$
Source: Page no :$3$
Theorem $1$:
Let $E/F$ be a transcendental extension with transcendence basis $S$. Then $E$ is algebraic over $F(S).$
Proof:Let $a\in E$ be arbitrary. If $a \in S$, then $a$ is a ...
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Set of representatives vs Teichmüller representatives
Say $K$ is a local field (complete wrt a discrete valuation with finite residue field, or if you want perfect residue field). Denote its residue field by $k$ where $k= \mathcal{O}_K / \mathcal{M}_K.$ ...
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Prove that $R / J$ is a field and determine $|R / J|$ with $J=(1-2 i) \subseteq \mathbb{Z}[i] .$
Exam question: In the ring of Gaussian integers $\mathbb{Z}[i]$, consider the ideal
$J=(1-2 i) \subseteq \mathbb{Z}[i] .$
(a) Prove that $R / J$ is a field and determine $|R / J|$.
(Check: The order $|...
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Properties of field extensions of finite transcendence degree
Let $K/F$ be a field extension of transcendence degree $n\ge 2$. If $E$ is a subfield of $K$ such that $F\subset E \subset K$ and the transcendence degree of $E$ over $F$ equals $n-1$,
then $K$ is a ...
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$f$ and $Df$ are not relative primes in $F[X]$, then they are not relative primes in $K[X]$.
The question I will ask originates in the context of the theory of perfect fields and separable extensions, but it is a question of irreducibility of polynomials between extensions of fields and of ...
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Using Eisenstein's Criterion with a transformation
I'm trying to prove that the following polynomial is irreducible in $\mathbb{Q}$ :$$14x^{10} + 18x^9 + 4x^3 + 1$$
Obviously, we can't apply Eisenstein's Criterion here so I tried setting $y = \frac{1}{...
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Every field extension generated by elements of degree two is normal.
Today I found an exercise that asked to demonstrate that every field $F/K$ extension generated by elements of degree 2 is normal.
If the extension were finitely generated, let's say $F=K(\alpha_1,\...
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Confused about Proof of Theorem regarding equivalent conditions for a finite extension E/F with Galois Group G.
A Theorem in my textbook says the following: The following conditions are equivalent for a finite extension $E/F$. with Galois group $G=Gal(E/F)$:
(i) $F=E^G$ (where $E^G$ are the elements of $E$ ...
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Splitting field of $X^{p^n}-1$ over $\mathbb{F}_p$
We know from Moore's theorem and the construction of finite fields that $\mathbb{F}_{p^n}$ is the splitting field of $X^{p^n}-X$ over $\mathbb{F}_p$.
I was wondering what the $X^{p^n}-1$ splitting ...
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Function field over a perfect field can be generated by two elements
I have two questions about the following theorem:
Theorem: Let $K$ be a perfect field, $F$ a function field in one variable over $K$ (i.e., a finite algebraic extension of $K(t)$). Then there is $x \...
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Question about finite extensions
A hint to an exercise in my textbook says "Since $B/F$ is finite, it is algebraic, and there are elements $α_1,...,α_n$ with $B=F(α_1,...,α_n)$."
I understand that if $B/F$ is a finite ...
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Is the converse of $\bmod p$ method in determining Galois group true? Or is there a prime corresponding to every member of the Galois group?
We have the following mod $p$ technique to detemine the Galois group of a polynomial in $\mathbb{Z}[x]$:
Assume $f\in\mathbb{Z}[x]$ is irreducible, $p$ is a prime,
and $\bar{f}$ is the polynomial ...
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Explicit Upper Bound for Class Number of Quadratic Field
I'm trying to find an explicit upper bound bound (explicit => without big O or small o notation, and with explicitly calculated constants) for the class number of a quadratic field.
So far, I've ...
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Proof of a telescoping formula for separable degrees in Hungerford's Algebra.
A corollary on page 287 of Hungerford's Algebra is
Corollary 6.13. If $F$ is an extension field of $E$ and $E$ is an extension field of $K$, then $$[F:E]_s[E:K]_s=[F:K]_s\mbox{ and } [F:E]_i[E:K]_i=[...
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Determening the field of invariants of some automorphisms of $\mathbb{Q}(x)$
Let $L=\mathbb{Q}(x)$ be the field of rational functions over $\mathbb{Q}$ and let $\sigma,\tau\in\text{Aut}(L)$ be given by $\sigma(f(x))=f(1/x)$ and $\tau(f(x))=f(1-x)$ for $f(x)\in\mathbb{Q}(x)$.
...
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Primitive element of a finite field whose powers do not lie inside the prime subfield
Let $p$ be a prime, and consider the finite field $\mathbb{F}_p$. Fix any $n\ge1$, and consider the field extension $\mathbb{F}_{p^n}/\mathbb{F}_p$. If $\alpha\in\mathbb{F}_{p^n}$ is a ...
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Why $[\mathbb{F}_{p}(\alpha): \mathbb{F}_{p^n}] = p$?
I was reading the second answer of the following question here Why $x^{p^n}-x+1$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$ :
Prove that $f(X) = X^{p^n} - X + 1$ is irreducible over ...
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Minimal extension of $\mathbb{Q}$ such that a polynomial split into conjugate factors
Let $p(X)\in \mathbb Z[X]$ be a given integral polynomial with no real zeros. The roots of $p$ comes in conjugate pairs, so we can write $p(X)=q(X)q^*(X)$ in $\mathbb C[X]$, such that $q$ is the ...
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Aluffi Alg. Chapter 0 Exercise 7.1.3
The first part of this exercise seems straightforward to me, but I'm a bit confused about the second part. In the case $\alpha$ is transcendental we know that $k(\alpha)$ is actually isomorphic to $k(...
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small side question (field theory, galois groups)
why does $\mathbb{Q}(3^{1/8},\zeta)=\mathbb{Q}(3^{1/8},2^{1/2}, i)$?
I know that $\mathbb{Q}(3^{1/8},\zeta)=\mathbb{Q}(3^{1/8},3^{1/8}e^{2\pi/8},3^{1/8}e^{4\pi/8},3^{1/8}e^{6\pi/8},3^{1/8}e^{8\pi/8},3^...
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Continuous automorphisms of number fields
Let $K$ be a Galois extension of $\mathbb{Q}$. Is it possible for non-trivial elements in $\text{Gal}(K/\mathbb{Q})$ to be continuous everywhere on $K$? Yes -- for instance, if $K$ is not a real ...
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Completing a proof that about field extensions
I already tried to ask this question, but it was considered a duplicate. I didn't find the associated post very explanatory, and probably I expressed myself poorly in the previous question. I try to ...
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Is it true that $i \in \mathbb{Q}(\sqrt{2},\xi)$, where $\xi = 1_{\frac{2\pi}{3}} = -\frac{1}{2}+i\frac{\sqrt{3}}{2}$?
I am trying to do the following exercise:
Give the Galois group of $\mathbb{Q}(\sqrt{3},\xi)$ over $\mathbb{Q}$, where $\xi = 1_{\frac{2\pi}{3}}$. Prove that $i \in \mathbb{Q}(\sqrt{2},\xi)$ and group ...
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Let $F$ be a field of characteristic $2$. Find the maximal separable subextension in $F(X)/F(X^4 + X^2)$.
Let $F$ be a field of characteristic $2$. Find the maximal separable subextension in $F(X)/F(X^4 + X^2)$.
I am not sure what to do here. I know that if $f(X) = aX^3 + bX^2 + cX + d \in \mathbb{F}_2[X]...
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Normal closure of $\mathbb{Q}(\sqrt{2 + \sqrt{2}})/\mathbb{Q}$
I am studying field theory and I am trying to find the normal closure of $\mathbb{Q}(\sqrt{2 + \sqrt{2}})/\mathbb{Q}$.
I know that normal extension (i.e. an algebraic extension in which every ...
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