# 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.

58 questions
1answer
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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, ...
1answer
85 views

### 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 ...
3answers
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0answers
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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 ...
1answer
41 views

### 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 ...
1answer
125 views

### 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 ...
0answers
65 views

### 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 ...
0answers
115 views

### 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 ...
1answer
115 views

### 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. ...
0answers
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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 ...
0answers
41 views

1answer
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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 ...
0answers
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### Finding the smallest field containing multivariate polynomial evaluations of the roots of an irreducible polynomial

$\newcommand\Q{\mathbb Q}$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 ...
1answer
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### (Real) Algebraic numbers that aren't constructible or roots

The third root of 2 is a real algebraic number. It is not constructible. If we had a set generated from all the constructible numbers and all finite roots (not just square roots) would this set be ...
2answers
495 views

### Is the Axiom of Choice needed for a Vitali set of algebraic numbers?

If we define a relation $\sim$ between real numbers so that $x \sim y$ holds precisely if $y - x$ is rational, then we need AC to prove that there exists a set of distinct representatives for the ...
1answer
66 views

### Why is the discriminant of a number field well-defined?

Let $K$ be a number field of degree $n$ and $z_1,\dots,z_n$ a $\mathbb{Z}$-basis for $\mathcal{O}_K$. It is clear that we can find another basis $y_1,\dots,y_n$ so, why $\Delta_K$ is independent of ...
0answers
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2answers
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3answers
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### Finding minimal polynomial over $\mathbb{Q}$

We note $m(x)=(x^2-2)^2-3$. $m(x)$ is a polynomial vanishing at $\alpha=\sqrt{2+\sqrt{3}}$. But how to prove that $m(x)$ is the minimal polynomial of $\alpha$ over $\mathbb{Q}$? Thanks