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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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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, ...
3
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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 ...
3
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3answers
48 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}...
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1answer
58 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 ...
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5 views

If $k(x,y) = \sum_r^n(\prod_{k\not =r}^n g_k(x) )*(f_r(x)) y^r$ is nonzero in the powers of $y$, is it nonzero in the powers of x also?

Let $u,v \in F/K$, a field extension of $K$ s.t $v$ is transcendental over $K$, and assume we know that for $$q(x) = \sum_r^n (\prod_{k\not =r}^n g_k(u) )* (f_r(u) x^r \in K(u)[x],$$ $q(v) = 0$, where ...
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1answer
20 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)(...
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1answer
56 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 ...
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1answer
40 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),...
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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)^...
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0answers
45 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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0answers
41 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
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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 ...
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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_{...
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1answer
116 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 ...
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3answers
78 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|+\...
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1answer
68 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 ...
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0answers
31 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
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1answer
65 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 ...
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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 ...
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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 ...
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0answers
24 views

From an arbitrary element in a number field find linear combination with respect to basis

I have an algebraic number $\alpha$ (a rough numeric approximation) over a number field $\mathbb Q(\theta)$. But I do not know how to express $\alpha$ as a linear combination of $1,\theta,\theta^2,\...
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2answers
87 views

Is it possible to find irreducible polynomial of this form..

I am searching for a sextic polynomial $f(x)$ that is irreducible over $\mathbb Q$ and factors in the following way over $\mathbb R$ $f(x)=((x+a)^2+d^2)((x+b)^2+d^2)((x+c)^2+d^2)$ where $d$ is a non-...
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3answers
68 views

How to check if $\sin\left(\frac{\pi}{3}\right)+\cos\left(\frac{\pi}{4}\right)$ is algebraic

Prove that $\sin\left(\frac{\pi}{3}\right)+\cos\left(\frac{\pi}{4}\right)$ is algebraic. Evaluating value of this, it is sum of two irrational numbers. How to find if it is algebraic?
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2answers
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Can the roots of a quadratic equation ever be a square root plus another square root?

Can one of the roots of a quadratic equation ever be of the form $\sqrt{x} + \sqrt{y}$? Assuming $x$ and $y$ are not perfect squares. The coefficients and constant of the quadratic equation need to ...
3
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1answer
99 views

Is every algebraic number smallest root of a polynomial with integer (or rational) coefficients?

Question: If $x$ is an algebraic number, then it occurs as a root of some polynomial with integer (or rational) coefficients. Is there also a polynomial with integer (rational) coefficients such that $...
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2answers
59 views

determine the degree of $2+\sqrt3$ over $\Bbb Q$

By using the binomial theorem for any $n\in\Bbb N$ $$(2+\sqrt3)^n=2^n+{n\choose1}2^{n-1}\sqrt3+{n\choose2}2^{n-2}3+{n\choose3}2^{n-3}3^{3/2}+...+{n\choose{n-1}}2\cdot3^{{n-1}\over2}+3^{n/2}\notin\Bbb ...
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2answers
49 views

Polynomial over a field whose power has coefficients in the ring of algebraic integers.

I have proven the result that given a number field $K$, and a monic polynomial $f \in \mathcal O_K\left[x\right]$ where $\mathcal O_K$ is the ring of algebraic integers in $K$, then any root of $f$ ...
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1answer
127 views

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 ...
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1answer
64 views

$\sin(2\pi q)$ is algebraic over $\mathbb{Q}$ [duplicate]

Let $q$ be an element of $\mathbb{Q}$ (rational numbers). How can I prove that $\sin (2\pi q)$ is algebraic over $\mathbb{Q}$ for any $q$? I am trying the method: Euler formula: $e^{i\theta} =\cos \...
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3answers
78 views

proof verification $\frac{3+2\sqrt{6}}{1-\sqrt{6}}$ is an algebraic integer

Is $$\frac{3+2\sqrt{6}}{1-\sqrt{6}}$$ an algebraic integer? An algebraic integer means an algebraic number in some algebraic number field $K\supset \Bbb Q$ that is the root of a monic polynomial $f\...
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2answers
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To find the minimal polynomial and to show that (1-i) is an associate of (1+i) in given set

I have two questions here:- -Find the minimal polynomial of $$\frac{(1+\sqrt[3]{7})}{2}$$ -Show that $(1-i)$ is an associate of $(1+i)$ in $\mathbb{Q}(i)$ For the minimal polynomial I guess I ...
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1answer
73 views

Once of the conjugates of an algebraic integer must have absolute value $\geq$ 1

Let $0\neq\omega \in \mathcal{O}_K$ be an algebraic integer. Prove that one of its conjugates has absolute value $\geq$ 1. My Thoughts: We know that the norm and trace of $\omega$ are in $\mathbb{Z}$...
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1answer
45 views

The maximal real algebraic field

I want to know what is the maximal real subfield of $\overline{\mathbb{Q}}$. Namely, what is $$\overline{\mathbb{Q}}\cap\mathbb{R}?$$ For a moment i thought that this was the field of totally real ...
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1answer
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Super Algebraic Numbers? [closed]

Let $P$ be a non-zero polynomial with real algebraic coefficients; prove or disprove the following "All real roots of $P$ are algebraic numbers"
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3answers
81 views

Is $5^{1/5} - 3\cdot i$ algebraic?

I am studying the book Complex Variables with Applications written by Herb Silverman. In this book, problem number 8 in Question 1.7 is as in the following. Is $5^{1/5} - 3\cdot i$ algebraic? (i.e, ...
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4answers
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If $\mathcal{O}_{\Bbb{Q}(\sqrt{d})}$ has class number $2$ or higher, does that mean $\sqrt{d}$ is irreducible but not prime?

Actually, I do know of exceptions to this, that's when $d$ is prime in $\Bbb{Z}$, e.g., $\Bbb{Z}[\sqrt{-79}]$ has class number $5$ and $\mathcal{O}_{\Bbb{Q}(\sqrt{79})}$ has class number $3$, but ...
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3answers
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Polynomial root of algebraic number

According to Wikipedia, an algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial in one variable with rational coefficients. The polynomial has many ...
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1answer
61 views

Is it true that evaluating a polynomial with integer coefficients at $e$, uniquely defines it?

In other words, there can not be two different polynomials with integer coefficients that evaluate to the same real number when $x=e$.
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2answers
50 views

Extension of field, $\Bbb{R}(i \pi) = \Bbb{C} $ [closed]

$K$ it's a field. Let $L|K$ ($L$ is extension of $K$), $a\in L$. If $a$ is an algebraic with degree $n$ over $K$, then a set $\{1,a,a^2,...,a^{n-1}\}$ is a basis of $K(a)$ over $K$. Also $K(a) = \{a_0 ...
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3answers
42 views

If $c\in\mathbb C$ is an algebraic number then for every $k\in\mathbb Z$, $kc$ is also algebraic

Prove that if $c\in\mathbb C$ is an algebric number then for every $k\in\mathbb Z$, $kc$ is also algebric So if $c$ is happened to be a rational it's trivial. How to solve the case of $c\in\mathbb ...
3
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2answers
86 views

How to show $1+\sqrt2 +\cdots+ \sqrt{2^n}$ is algebraic?

How to show that $1+\sqrt2 +\cdots+ \sqrt{2^n}$ is an algebraic? $x=1+\sqrt2 +\cdots+ \sqrt{2^n}$ $x-1=\sqrt2 +\cdots+ \sqrt{2^n}$ Every other element is an integer so I can move it to the left ...
2
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1answer
152 views

What is an Algebraic function field?

I can't understand what is an Algebraic function field. Some references says that, $F/K$ is an Algebraic function field if it contains at least one transcendental number. I found and example for a ...
4
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1answer
80 views

Algebraic numbers on both sides of trigonometric functions

In this answer I learned about Niven's theorem. As I understand it, it says $$\left\{t\in\pi\mathbb Q\mid 0\le t\le\frac\pi2\wedge\sin(t)\in\mathbb Q\right\} =\left\{0,\frac\pi6,\frac\pi2\right\}$$ ...
0
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3answers
163 views

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