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.
13,723 questions
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If $K/F$ and $L/F$ are separable, then the composite $KL/F$ is separable
Prove that, if $K/F$ and $L/F$ are separable, then the composite $KL/F$ is separable.
My attempt: for finite extensions, this useful lemma holds:
Let $K/F$ be a field extension. If $\alpha_1, \dots, ...
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Splitting field of a polynomial over $\mathbb{Q}_p$?
How to find the degree and the ramification index of the splitting field of a certain polynomial over $\mathbb{Q}_p$? For instance, $f(x)=3+9x+3x^4+x^6$ over $\mathbb{Q}_3$?
If $\gamma$ is one of the ...
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Prove that the extension $\mathbb{Q} \subset \mathbb{Q}(\sqrt{3 + \sqrt{7}})$ is not normal
I want to prove that the extension $\mathbb{Q} \subset \mathbb{Q}(\sqrt{3 + \sqrt{7}}) = L$ is not normal. My strategy is to show that the minimal polynomial $f(x) = x^4 - 6x^2 + 2$ of $\alpha = \sqrt{...
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Prove $x^{46}+69x+2025$ is irreducible in $\mathbb Z[x]$
I was told to work in $\mathbb F_{23}$, and also show it has a linear factor $\mathbb Z_5$
Write $f(x)=x^{46}+69x+2025$. We begin by supposing that $f=gh$ for some $g,h \in \mathbb Z[x]$. First, $g$ ...
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Not all quadratic extensions over $\mathbb{Q}$ are contained in the compositum of all the splitting fields of irreducible cubics in $\mathbb{Q}[X]$
Question: Let $F$ be the composite of all the splitting fields of irreducible cubics over $\mathbb{Q}$. Prove that
$F$ does not contain all quadratic extensions of $\mathbb{Q}$.
(This is exercise 16 ...
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Are these sequences multiplicative?
OEIS sequences A352550 to A352560 are of the form "$a(n)=$ number of modules with $n$ elements over the ring of integers in the real quadratic field of discriminant $d$", and A352561 to ...
- 473k
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Why two minimal polynomials of the algebraic numbers that define a composite algebraic field do not share roots? [closed]
I am reading Galois Theory from Postnikov. In the first chapter, he tries to prove that every composite algebraic extension of $P$, that is $P\subset P(\alpha_1)\subset K=P(\alpha_1)(\alpha_2)$ (in ...
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Quadratic fields in $\mathbb{Q}(p^{\frac{1}{n}})$ where $n$ is even,p is prime.
I want to prove that there is a unique quadratic field extension in $\mathbb{Q}(p^{\frac{1}{n}})$, where $n = 2k$ and p is prime. Clearly $\mathbb{Q}(p^{\frac{1}{2}})$ $\subset$ $\mathbb{Q}(p^{\frac{1}...
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Can the integral closure of a ring be taken intrinsically?
If $A \subset B$, both commutative rings, then we define $\bar{A}_B = \{x \in B| b \text{ is integral over A}\}$ where $x$ being integral over $A$ means there is a monic relation $a_0 + a_1x + \ldots ...
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The degree of division in non-integral domain
Let $F$ be a finite field. We define the algebra $J=F[x] / (x^{p}-1)$ where $p$ is any prime. Let`s consider the elements $g(i,j)\in J$, $g(i,j)=(x^{-1}-1)^i(\sum_{s=0}^{p^2}x^{s}s^{p-2})^j$ for $i,j\...
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Is there a canonical choice of algebraic closures for finite fields?
From what I understand, there is an almost canonical choice for an algebraic closure of $\mathbb{Q}$. Namely, take all algebraic elements in $\mathbb{C}$. As the extension $\mathbb{R}/\mathbb{Q}$ is ...
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Finite axiomatization for the first-order theory of the multiplicative reducts of fields.
Consider the class $C$ of algebras of type $<2,0,0>$ which are the $(*,0,1)$ reducts of fields. I want to know if there is a finite set of axioms that determine the first-order theory $Th(C)$ of ...
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The composite of fields $L_1 L_2$ where $L_1,L_2$ are finite over $F$ is the set of finite sums of products $ab$ with $a \in L_1$ and $b \in L_2$
Let $L_1$ and $L_2$ be field extensions of $F$ contained in a larger field $K$. If $K/F$ is finite, we define the composite $L_1 L_2$ to be the smallest subfield of $K$ containing $L_1$ and $L_2$. ...
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If $p=8n+3$ is a prime then $f=t^2+2$ splits over $F_p$.
If $p=8n+3$ is a prime then $f=t^2+2$ splits over $F_p$.
I’d like if someone could guide me through an idea to prove it using field/Galois theory.
The only idea I have so far is to assume $f$ doesn’...
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Choosing a desired embedding of $\overline{\mathbb{Q}}$ in $\overline{\mathbb{Q}}_p$
Suppose I have a local field $L/\mathbb{Q}_p$ with prime $\mathfrak{p}$. I want to take an embedding $\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_p$ so that I can see $G(\overline{\...
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Interpretation of the formula for the number of generators of a finite field extension
Let's consider the finite field extension $\mathbb{F}_q \subseteq \mathbb{F}_{q^n}$. Using Möbius inversion, one can get that the number of generators of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$ is $$\...
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Absolute Discriminant and analogy of genus with function fields
In 2.6 of https://math.mit.edu/~poonen/papers/curves.pdf, Poonen has a table of analogous concepts between number fields and function fields.
Some of these analogies are not too unfamiliar to me but ...
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reciprocal of $\sqrt[3]{M}-\frac{1}{\sqrt[3]{M}}-a$ [duplicate]
Let
$$
x=\sqrt[3]{M}-\frac{1}{\sqrt[3]{M}}+a
$$
where $a\in\mathbb Q$ and $M$ is an integer such that
$\sqrt[3]{M}\notin\mathbb Q$. Note that $x\in\mathbb Q(\sqrt[3]{M})$. Clearly $\frac{1}{x}\in\...
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Understanding transcendence degree and transcendence basis of domains
The following definition is taken from Kemper's commutative algebra textbook.
Let $A$ be an algebra over a field $k$. Then the transcendence degree of $A$ is defined as
$$\operatorname{trdeg}(A) := \...
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Bound on the degree over $k$ of a field extension $k\subseteq L \subseteq M_n(k)$
Let $k$ be a field, and $n$ be a positive integer. Then $k$ is embedded inside $M_n(k)$ (the set of $n×n$ matrices over $k$) as the set of scalar matrices $k'=\{aI_n:a \in k\}$.
Suppose $L\subseteq ...
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Is there an infinite non-orderable field that can be partitioned in this way?
This is a follow-up to my previous question, here: Can the complex numbers be partitioned in this way?. Refer to the previous question for the definition of anti-closure. Anyway, I have noticed that ...
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The set of algebraic numbers with odd degrees appears to form a field
I'm exploring whether the roots of all irreducible polynomials with odd-degree integer coefficients form a field. It seems they don’t, because the sum or product of two such roots may not have an odd-...
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Necessity of proving addition to both sides of an equation (axiomatic construction of Q)
I am beginning my first course in real analysis. We have begun with an axiomatic construction of the rational numbers. I've included our particular set of axioms at the end for reference.
My professor ...
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Algorithm for minimizing the degree of a vector of algebraic numbers
Given a vector of algebraic numbers, $\vec{a} = (a_1, a_2, \dots, a_n)$, let the "max algebraic degree" be $$
\operatorname{maxDeg}(a_1, a_2, \dots, a_n) = \max(\deg(a_1), \deg(a_2), \dots, \...
- 5,438
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Show that the minimal polynomial of $\sqrt{3}+\sqrt{5}$ over $\mathbb{Q}$ is irreducible in $\mathbb{Z}[t]$ but becomes reducible modulo any prime $p$
Problem: Find the minimal polynomial of $\sqrt{3}+\sqrt{5}$ over $\mathbb{Q}$ and show that it is an irreducible polynomial in $\mathbb{Z}[t]$ which becomes reducible modulo any prime $p$.
I found ...
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