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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2
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2answers
21 views

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 ...
1
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1answer
33 views

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 ...
2
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2answers
122 views

Find the polynomial of integral coefficient with minimum degree and root $z+z^3+z^9$.

Let $z$ be a 13th root of unity $(z\neq 1)$. Find the polynomial of integral coefficient with minimum degree and root $z+z^3+z^9$. My idea: since $z$ such $$0=z^{12}+z^{11}+\cdots+z+1=\prod_{k=1}^{...
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0answers
24 views

For which regular $n$-gons are there two sides/diagonals with a rational ratio other than $1$?

Context: I'm trying to see which regular polygons can be built using points generated from a wallpaper group. I know that if a regular polygon has more than a certain amount of these vertices, some ...
1
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0answers
54 views

Is $x^0$ a polynomial?

On the one hand I have seen this called a "constant polynomial" since $x^0 = 1$ for any $x$. But I would have thought this was a bad idea, because then any number can be considered the root of the ...
0
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1answer
41 views

algebraic number $e^{\pi i /15}$

How do I know that $$e^{\pi i /15}$$ is algebraic number? I know it is from the worlfram, but I don't know what method I can use to find if a number is algebraic.
0
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0answers
20 views

A decimal Bailey–Borwein–Plouffe type formula for the Bring radical of 1?

The Bailey-Borwein-Plouffe formula yields a hexadecimal spigot algorithm for the mathematical constant $\frac{\tau}{2} = \pi$. Does there exist a decimal (base 10) Bailey–Borwein–Plouffe type formula ...
0
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0answers
78 views

Solutions to Marcus - Number Fields

I'm an undergrad mathematics student and have just read Number Fields by Marcus. I have searched for a while for a complete set of solutions but only seem to be able to find solutions to Chpt 1 and 2. ...
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0answers
20 views

Bit complexity of computing the sign of an expression evaluated at an algebraic number

I have a univariate polynomial $F(t)\in \mathbb{Z}[t]$ of degree $d$ and maximum bitsize of coefficients equal to $\tau$ and $G(t) \in \mathbb{Z}[t]$ of degree $d^2$ and maximum bitsize of ...
11
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2answers
396 views

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 ...
0
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1answer
20 views

Proving equivalence between finite field extensions.

Given $K=\mathbb{Q}(\omega)$ where $\omega$ is a complex cube root of 1. The minimal polynomial of $\omega$ is of degree 2 and we have that $\tau = (a+b\omega)\in K$ ($a,b\in \mathbb{Q}$) has minimal ...
1
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1answer
43 views

Algebraic degree of $\cos\left(\frac{p\pi}{q}\right)$

How can we find the algebraic degree of $\cos\left(\frac{p\pi}{q}\right)$ for $p$, $q$ coprime integers? I know that the algebraic degree of $e^{\frac{p\pi i}{q}}$ is $\phi(q)$, since cyclotomic ...
1
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0answers
47 views

Discriminant of a basis

I am trying to find the discriminant of the Q-basis $ ( 1,\sqrt[3]{3},\sqrt[3]{9}) $ . I want to do it by explicit matrix computations. I know that I have to find embeddings, but I think there is only ...
2
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1answer
69 views

Show that if $\beta \in E$ is algebraic over $F(\alpha)$, then there is a nonzero polynomial $f(x,y) \in F[x,y]$ such that $f(\alpha, \beta)=0$.

Let $E$ be an extension field of F, and let $\alpha \in E$ be transcendental over $F$. Show that if $\beta \in E$ is algebraic over $F(\alpha)$, then there is a nonzero polynomial $f(x,y) \in F[x,y]$ ...
1
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1answer
78 views

Trace of polynomials/algebraic integers

I am somewhat stuck with the proof of Lemma 2.3 on page 6 in this paper about an algorithm for polynomial factorization. The preliminaries: We have $f \in Z[X]$ monic, squarefree (which here means ...
0
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2answers
76 views

Determining algebraicity of $\cos1^\circ$ and $\sin1^\circ$

Let $a = \cos1^{\circ} $ and $b = \sin 1^{\circ}$ We say that a real number is algebraic if it is a root of a polynomial with integer coefficients. Then- A. $a$ is algebraic but $b$ ...
0
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1answer
49 views

Given a minimum polynomial for $\alpha$, find the minimum polynomial for $\alpha/3$

Let $f(x)=x^3+2x^2+x+27$. Suppose $\alpha\in\mathbb{C}$ satisfies $f(\alpha)=0$, and let $K=\mathbb{Q}(\alpha)$. Is $f(x)$ irreducible? What is $[K:\mathbb{Q}]$? What is $N_K(\alpha)$? Is $\frac{\...
4
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1answer
180 views

Find the discriminant of $\mathbb{Q}(\sqrt{3},\sqrt{5})$

So far I have that $B = \{1,\sqrt{3},\sqrt{5},\sqrt{3}\sqrt{5}\}$ is a $\mathbb{Q}$-basis for $\mathbb{Q}(\sqrt{3},\sqrt{5})$. I think the discriminant of $B$ is $2^83^25^2$, which implies that, if $...
3
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1answer
49 views

Is there terminology for these rings of algebraic integers of degree $4$?

As you already know, given a squarefree integer $d > 1$, the ring of algebraic integers of $\textbf Q(\sqrt d)$ contains irrational algebraic integers of degree $2$ but is purely real. Given ...
2
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2answers
66 views

if the sum of two numbers $\alpha$ and $\beta$ is algebraic, and their product is transcendental, what do we know about these numbers?

These are elements of a field. My intuition says that $\alpha=a+b$, $\beta=a-b$, where, $a$ is algebraic and $b$ is transcendental, but I can't prove it. I don't even know where to start. Thanks in ...
7
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2answers
139 views

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 ...
7
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1answer
157 views

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, ...
10
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5answers
425 views

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 MSE: $$-\tan\frac{\pi}{5}+4\sin\frac{2\pi}{5}=\tan\frac{\pi}{5}+4\sin\frac{\pi}{5}=\sqrt{5+2\sqrt{5}}$$ $$-\tan\frac{2\pi}{7}+4\sin\...
2
votes
1answer
139 views

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 ...
1
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1answer
155 views

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 ...
1
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1answer
64 views

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 ...
1
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1answer
64 views

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, ...
3
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2answers
126 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 ...
4
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3answers
71 views

$\sqrt[3]{3} + \sqrt[3]{9}$ is algebraic over $\mathbb{Q}$

Show that $\alpha = \sqrt[3]{3} + \sqrt[3]{9}$ is algebraic over $\mathbb{Q}$ by presenting a polynomial $p$ from $\mathbb{Q}[X]$ with $p(\alpha)$. It seems like $$\mathbb{Q}(\sqrt[3]{3} + \sqrt[3]{9}...
2
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1answer
68 views

If $x$ and $y$ are complex numbers and $x+y$ , $xy$ are algebraic numbers then how to prove that $x$ and $y$ are also algebraic numbers?

I tries basic operations like multiplication and addition in a hope that i will get $x$ and $y$ out of $x+y$ and $xy$ but that didn't worked for me.Also i tried assuming a polynomial with rational ...
1
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1answer
29 views

Degree of $\cos(2\pi/8) + i \sin(2\pi/8)$ over $\mathbb{Q}$

What is the degree of $\cos(2\pi/8) + i \sin(2\pi/8)$ over $\mathbb{Q}$ ? I note that $\cos(2\pi/8) + i \sin(2\pi/8)$ is a root of $x^8-1$. $x^8-1$ can be factored into $x^8-1 =(x^4+1)(x^2+1)(x+1)(...
0
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1answer
165 views

Show $\cos(\pi/11)$ is algebraic over $\mathbb{Q}$

Show $\cos(\pi/11)$ is algebraic over $\mathbb{Q}$ I am trying to follow the answers in How to prove that $\cos (2\pi/n)$ is algebraic? So $\cos(\pi/11)+ i\sin(\pi/11)$ is a root of $x^{22}=1$ and ...
1
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1answer
57 views

Are Galois conjugates of a prime of a cyclotomic ring also primes?

For the sake of simplicity consider $\mathbb{Q}[\zeta_{5}]$. If a cyclotomic integer $z\in\mathbb{Z}[\zeta_{5}]$ is a prime of the integer ring, is it true that its Galois conjugates $\{z, \sigma_1(z),...
12
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3answers
3k views

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)^...
2
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0answers
47 views

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 ...
1
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1answer
42 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 ...
0
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1answer
169 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 ...
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0answers
75 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 ...
20
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3answers
258 views

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 ...
2
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0answers
150 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 ...
6
votes
1answer
122 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. ...
1
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0answers
29 views

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 ...
1
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0answers
44 views

Minimizing the degree of a set of real algebraic numbers

Motivation If you were asked to write down the coordinates of a set of four points in $\mathbb{R}^3$ that form a regular tetrahedron, you might come up with $$ \left\{(0, 0, 0),(1,0,0),\left(\frac{1}...
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0answers
33 views

Is $K$ here a subfield of $\Bbb C$? [duplicate]

In Jürgen Neukirch's "Algebraic Number Theory", page 5. An algebraic number field is a finite field extension $K$ of $\Bbb Q$. The elements of $K$ are called algebraic numbers. Is $K$ here a ...
0
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1answer
48 views

Is the number $\sum_{k \in C}\frac{1}{p^k}$ an algebraic number? [closed]

Let set $C$: $C \subset \mathbb{Z}^+$, and give $c \in \mathbb{Z}^+, c > 1$. if $\sum_{k \in C}\frac{1}{c^k}$ is an algebraic number, for other $p \in \mathbb{Z}^+, p > 1$, is the number $\sum_{...
3
votes
1answer
391 views

a way to represent algebraic numbers in a computer

Say you want to represent the rational numbers in a computer. This is quite easy, you can think of them as pairs of integers. It is also easy to develop algorithms for adding, subtracting, multiplying ...
0
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3answers
96 views

On algebraic numbers

Walter Rudin Exercise 2.2 To prove that the set of all algebraic numbers is countable, the hint provided is that there are finitely many equations of the form $$n+\left|a_0\right|+\left|a_1\right|+\...
3
votes
1answer
94 views

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 ...
0
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0answers
34 views

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 ...
2
votes
1answer
91 views

(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 ...