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.
195 questions
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Is the continued fraction $[3;1,4,1,5,9,…]$ known to be transcendental?
I became interested in the continued fraction whose partial quotients are the non-zero decimal digits of $[3;1,4,1,5,9,2,6,5,3,…]$.
Because the sequence never terminates, the value is certainly ...
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Is $A_n$ dense in $(0,\infty)$?
Let $A_n$ be the set of algebraic integers whose minimal polynomial has degree $n$ and all of its roots positive. Is $A_n$ dense in $(0,\infty)$ if $n > 1$?
Without the condition that all of its ...
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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 $$
\operatorname{maxDeg}(a_1, a_2, \dots, a_n) = \max(\deg(a_1), \deg(a_2), \dots, \...
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Is it more likely for an algebraic number to be irrational than rational?
A real number is more likely to be transcendental than algebraic, simply by cardinality considerations. However, suppose we are given that a real number $r$ is algebraic. Is it then more likely that $...
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What equivalence relation is implicit in the Kodaira symbol classification of Elliptic curves?
For an elliptic curve $E/\mathbb{Q}_p$, one takes its Neron minimal model. Let $C$ be its special fiber, which is $1$ dimensional scheme over $\bar{\Bbb{F}_p}$.Kodaira’s notation $(I_n, II, III,IV,I_n^...
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Infinite series of $1/9^{F_n}$ yields algebraic number?
This document contains a problem that reads as follows:
Let $F_n$ be the nth Fibonacci number defined by $F_1 = 1$ and $F_2 = 1$ and for all $n\geq 3$, $F_n = F_{n-1} + F_{n-2}$. Prove that $$\sum_{n=...
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Finding an integer-coefficient polynomial for $\sqrt[3]2+\sqrt[3]3$ [duplicate]
We need to find a polynomial with integer coefficients whose roots include $x=\sqrt[3]2+\sqrt[3]3$, by hand.
Here are some ideas, but none of them are easy enough by hand.
Write the system of ...
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Generator for a nasty number field
I have a collection of algebraic numbers, which I'll list along with their minimal polynomials:
\begin{array}{cc}
a_1 = \sqrt{\frac{1}{2} \left(1+2 \sqrt{2}+\sqrt{3}+\sqrt{6}\right)} & p_1(x)= 2 ...
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Is $(1 + \sqrt{17})/(2i\sqrt{19})$ an algebraic integer?
I'm learning the basics of algebraic number theory for the first time and I'm confused about a simple question: how can I decide if the number
$$\alpha = \frac{1 + \sqrt{17}}{2i\sqrt{19}}$$ is an ...
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A Question Regarding the Proof of the Lindemann Weierstrass Theorem
I am looking at a proof of the Lindemann Weierstrass Theorem: given $u_1, u_2, …, u_n, v_1, v_2, …, v_n $ algebraic numbers over $\mathbb{Q}$, with the $v_i$ distinct, if $u_1e^{v_1} + u_2e^{v_2} + … +...
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Is there any relation between the dihedral angles in the regular triangle (60 degrees), tetrahedron (70.53), pentachoron (75.52), etc.?
The angle in a regular $n$-dimensional simplex is $\theta_n=\arccos(1/n)$. Is there a (finite) list of integers $c_2,c_3,c_4,\cdots$ such that
$$c_2\theta_2+c_3\theta_3+c_4\theta_4+\cdots=0,$$
other ...
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Show that 1 and -1 are the only units in the ring of integers of $\mathbb{Q}[\sqrt{m}]$ for $m$ squarefree, negative, and not equal to -1 or -3.
For reference this is exercise 13 in chapter 2 of Marcus's book Number Fields.
For the sake of simplicity I'll work in the case $m \not\equiv 1 \mod 4$ so that the ring of integers of $\mathbb{Q}[\...
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Basis of a number field relative to an ideal of $\mathcal{O}_K$
I'm trying to understand part of the proof of the following proposition.
"If $K/\mathbb{Q}$ is a finite extension of degree n, then every nonzero ideal $I\subseteq\mathcal{O}_K$ contains a basis ...
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Prove that there is no largest algebraic number field
I stumbled upon this question: Rigorously prove that there does not exist a largest algebraic number field (a finite extension field of rational numbers) that contains all algebraic number field. Or ...
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How to prove that any number which can be written w/ the four operations and radicals on rationals are roots of polynomials with integer coefficients?
Few days ago I watched this (https://youtu.be/WyoH_vgiqXM?si=MkD03EApG5mwa5zu) amazing video by Mathologer on $e$ and $π$ transcendence proof. Then I remembered a question I have about $e$ and $π$: ...
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