# Questions tagged [algebraic-numbers]

Use this tag for questions related to numbers that are roots of a non-zero polynomial in one variable with rational coefficients.

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### Why can't $\alpha - 1$ be a unit here?

I've been reading a the book "Distribution Modulo one and Diophantine Approximation" by Yann Bugeaud. Bugeaud is proving a statement due to Toufic Zaïmi about Salem numbers $\alpha$. A Salem number is ...
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### Computing ramification in extension of complete DVRs

Assume I am given a finite primitive extension of complete discretely valued fields $L=K(\alpha)/K$, say with monic integral minimal polynomial $f$ for $\alpha$. How does one systematically compute ...
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### If $a$ and $b$ are algebraic, then $\frac ab$ is algebraic

Prove that if $a$ and $b$ are algebraic over field $\mathbb F$ and $b \neq 0$, then $\frac ab$ is algebraic over $\mathbb F$. I know that algebraic means that there are polynomials $f$, $g$ such that ...
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### Product of transcendental numbers is not transcendental, or is it?

The transcendental numbers form a field, or so I thought. I'm familiar with the fact that the algebraic numbers form a field which implies that reciprocals of transcendental numbers must be again ...
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### Sub-Sum of Roots of Unity

Let $\alpha$ be an algebraic integer of a cyclotomic field, and let $\theta_1, \theta_2, ..., \theta_n$ be roots of unity such that $$\sum_{i=1}^n \theta_i = 2\alpha.$$ Does there necessarily exists a ...
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### Polynomial with Root $\pi+ei$

How do I find a polynomial with a root of $\pi +ei$ in the reals? I know how to do this with algebraic numbers, but not transcendental ones like e and $\pi$. Edit: I now realize that in the reals, ...
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### A special inverse Galois problem

Suppose that $G$ is a transitive permutation group and suppose we know a construction of an isomorphism from the Galois group of a Galois number field to $G$. Does this information make it easier to ...
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### Integer representatives for $F^{\times} / N_{E/F}(E^{\times})$?

It's been a while since algebraic number theory, so I apologize if this is too simple. Let $F=\Bbb{Q}(\sqrt{s})$ and $E=F(i\sqrt{t})$, where $s,t > 0$ are square-free integers. I would like to ...
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### Is there 'Algebraic number' which cannot display with Arithmetic operation and root

Let $a + bi$ be an algebraic number. Then there is polynomial which coefficients are rational number and one of root is $a+bi$. I think.. $$x = a + bi$$ we can subtract $c_1$ (which is rational ...
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### Is “indeterminate” a synonym for “variable” or for “transcendent”?

The following is from Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra, Edited by H. Behnke, F. Bachmann, K. Fladt, W. Süss and H. Kunle. Bold text ...
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### How to interpret action of $SL_2(\mathcal{O}_d)$

Given a lattice $\wedge = \{\omega_1, \omega_2 \}$ in $\mathbb{C}$, $\omega_1 / \omega_2 \not\in \mathbb{R}$, we know that $\wedge' = \{\omega_1', \omega_2' \}$ defines the same lattice precisely when ...
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### Strictly increasing function from reals to reals which is never an algebraic number

Let $f:\Bbb R\rightarrow\Bbb R$ have the properties $\forall x,y\in\Bbb R,\space x<y\implies f(x)<f(y)$ and $\forall x\in\Bbb R,\space f(x)\notin\Bbb A$ where $\Bbb A$ is the set of algebraic ...
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### Are these enough to research in Algebraic Number Theory? [closed]

$\text{Algebraic Number Theory:}$ To do research in Algebraic Number Theory what are the essential topics to know? I have basic knowledge in Abstract Algebra, Topology and Analysis as well as the ...
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### Real irrational algebraic numbers “never repeat”

An oft-used phrase describing irrational numbers is that their (decimal) expansions "never repeat". The sense of "never repeating" intended is, of course, that their expansions don't repeat forever. ...
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### Sum of algebraic number of prime degree. [duplicate]

I'm looking to prove the following: Let $K$ be a field and suppose that $[K(\alpha):K]=p$ and $[K(\beta):K]=q$ for $p,q$ distinct primes. Then $K(\alpha+\beta):K$ has degree $pq$. Of course I know ...
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### are parabolic points in the Mandelbrot set algebraic numbers?

Define the iterated quadratic polynomial: \begin{aligned} f_c^0(z) &= 0 \\ f_c^{n+1}(z) &= f_c^n(z)^2+c \end{aligned} The $c$ for which $f_c^n(0)$ remains bounded form the Mandelbrot ...
$\newcommand\Q{\mathbb Q}$Suppose I have an irreducible polynomial $f\in\Q[x]$ and suppose I know the roots say $r_1,\dots,r_n\in \bar \Q$. I want to know if there is an easy way to "compute" the ...