Mathematics Stack Exchange
Mathematics Stack Exchange

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.

An algebraic number is any complex number that is a root of a non-zero polynomial in one variable with rational coefficients or, equivalently by clearing denominators, with integer coefficients. The set of all algebraic numbers is usually denoted by $\mathbb A$.

All integers and rational numbers are algebraic as are all roots of integers (including $\pm i$). The same is not true for all real and complex numbers because they also include transcendental numbers such as $\pi$ and $e$.

If $a$ and $b$ are algebraic numbers, then so are the numbers $a+b$, $-a$, $ab$ and (if $a\neq0$) $1/a$. Therefore, $\mathbb A$ is a field.

121 questions
81
votes
3 answers

Prove that the set of all algebraic numbers is countable

A complex number $z$ is said to be algebraic if there are integers $a_0, ..., a_n$, not all zero, such that $a_0z^n+a_1z^{n-1}+...+a_{n-1}z+a_n=0$. Prove that the set of all algebraic numbers is countable. The Hint is: For every positive integer…
PandaMan
  • 2,949
  • 2
  • 24
  • 44
51
votes
11 answers

What if $\pi$ was an algebraic number? (significance of algebraic numbers)

To be honest, I never really understood the importance of algebraic numbers. If we lived in an universe where $\pi$ was algebraic, would there be a palpable difference between that universe and ours? My choice of $\pi$ for this question isn't…
ante.ceperic
  • 4,911
  • 2
  • 23
  • 48
21
votes
3 answers

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
stanley dodds
  • 987
  • 1
  • 5
  • 13
14
votes
2 answers

Is there a general formula for $\sin( {p \over q} \pi)$?

Virtually everyone knows the basic values of the unit circle, $\sin(\pi) = 0; \ \ \sin({\pi \over 2}) = 1; \ \ \sin({\pi \over 3}) = {\sqrt{3} \over 2} \\$ And other values can be calculated through various identities, like $\sin({\pi \over 8})…
Rob Bland
  • 5,982
  • 13
  • 29
13
votes
5 answers

Trig identities analogous to $\tan(\pi/5)+4\sin(\pi/5)=\sqrt{5+2\sqrt{5}}$

The following trig identities have shown up in various questions on…
Franklin Pezzuti Dyer
  • 37,332
  • 9
  • 60
  • 145
12
votes
3 answers

How to divide one number in $\textbf Q(\zeta_8)$ by another?

Consider two numbers, one is $a + b \zeta_8 + ci + d(\zeta_8)^3$, the other is $\alpha + \beta \zeta_8 + \gamma i + \delta(\zeta_8)^3 \neq 0$. How do I compute $$\frac{a + b \zeta_8 + ci + d(\zeta_8)^3}{\alpha + \beta \zeta_8 + \gamma i +…
Bill Thomas
  • 516
  • 1
  • 6
  • 21
10
votes
2 answers

Proof that $\log_23 +\log_52$ is irrational number

Problem is to prove that $$\log_23 +\log_52$$ is irrational number. My attempt: I try to write number like $$\log_23 +\frac{1}{\log_25}$$ but I didn't get anything(proof by contradiction). I also try to find polynomial such that given number is…
josf
  • 1,277
  • 9
  • 25
8
votes
2 answers

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 equivalence classes of $\sim$. Do we also need AC to…
Tommy R. Jensen
  • 157
  • 9
8
votes
1 answer

How does knowing that $\sqrt{7}\notin\mathbb{Q}(e^{2\pi i/7})$ help construct this character table?

Here's a question that has haunted me since it appeared on a problem sheet in a Representation Theory course I attended as an undergraduate. I'll reproduce it exactly: A group of order $168$ has $6$ conjugacy classes. Three representations of this…
Oscar Cunningham
  • 15,389
  • 3
  • 43
  • 77
7
votes
2 answers

Why are algebraic numbers important and worth defining?

Yes, this is a soft question. Hold your horses though: I’ve met several criteria specified in How to ask a good question, so it does not warrant an “opinion-based” closure. Algebraic numbers are those numbers which are zeroes of polynomials with…
gen-ℤ ready to perish
  • 6,434
  • 3
  • 25
  • 41
7
votes
1 answer

How to round to algebraic integers in real quadratic integer domains

I feel like this question has been asked here before, but I'm not finding it. In an imaginary quadratic integer domain, it is very easy to round algebraic numbers to algebraic integers. For example, there are four choices for rounding $$\frac{1}{2}…
Bob Happ
  • 520
  • 1
  • 5
  • 23
6
votes
0 answers

Minimal polynomials of "simple" algebraic numbers

This should be a fairly trivial question, to which I have nevertheless found no satisfactory answer. I am interested in effective algorithmic computation of minimal polynomials of some particularly simple algebraic numbers, especially the complex…
Jára Cimrman
  • 522
  • 2
  • 12
6
votes
2 answers

Prove that the sum $ \sqrt{1001^2 + 1}+\sqrt{1002^2 + 1} \ + ... + \sqrt{2000^2 + 1}$ is irrational

Prove that the sum $$ \sqrt{1001^2 + 1}+\sqrt{1002^2 + 1} \ + ... + \sqrt{2000^2 + 1}$$ is irrational. The textbook has the solution too but I'm unable to understand it. The strategy is divided into two parts:- Proving that the sum is not an…
Aryamaan
  • 73
  • 5
6
votes
1 answer

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. And it's straight-forward to show that rational…
ThePopMachine
  • 161
  • 6
5
votes
0 answers

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 $$\begin{pmatrix} \omega_1' \\ \omega_2'…
Quotable
  • 631
  • 3
  • 12
1
2 3
…
8 9