1
vote
0answers
39 views

Series of polynomials and uniformly convergence

It's part of the proof of a Lemma of an article I was reading (Algebraic values of transcendental functions at algebraic points). I couldn't understanding one thing: Let f be a complex function such ...
6
votes
1answer
80 views

Is there a simple procedure to produce algebraic numbers of modulus one that are not roots of unity?

Let $z=e^{i\theta}$ be a complex number of modulus $1$. Trivially if $z$ is a root of unity then $z$ is also an algebraic number, but the converse is known to be false : $z$ can be algebraic without ...
0
votes
2answers
33 views

Show that $x$ is an algebraic number? Where $x$ is…

Can someone help me with the following problem? Show that $x=\sqrt2+\sqrt[3]3$ is an algebraic number. By finding a polynomial with rational coefficients for which $x$ is a root of. Can someone ...
1
vote
2answers
67 views

number theory of coefficients in an infinite sequence of polynomials

EDIT: equivalent formulation by Hurkyl in comments: if $n$ is odd and $p^\nu \parallel n$ and $n > 2k,$ then $$ p^{(\nu + 2 + 2 k - n)} \; | \; \sum_j \left( \begin{array}{c} n \\ 2j \end{array} ...
2
votes
0answers
33 views

The existence of Pisot numbers in any real number field

Wikipedia claims that, given a real algebraic number field $K$ of degree $n$, there is an algebraic integer $r \in K$ of degree $n$ such that $r>1$, but every conjugate of $r$ has modulus $<1$ ...
6
votes
2answers
129 views

Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?

We know that $f(x)=x^4+1$ is a polynomial irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$. My question is: Is it true that if $f(x)$ has a linear factor over ...
6
votes
2answers
76 views

Checking irreducibility of polynomials over number fields

Are there general methods for checking irreducibility of polynomials over number fields? For instance, letting $F = \mathbb{Q}(\sqrt{3})$, I want to know whether $x^3 - 10 + 6\sqrt{3}$ is irreducible ...
1
vote
2answers
112 views

Finding number of positive integral solutions of $x^4-y^4=3879108$

Find the number of positive integral solutions of $$x^4-y^4=3879108$$ $$3879108=36*277*389$$ I tried simplifying factors of $3879108$ to get terms in the form of $x^4-y^4$. However, I am unable to ...
4
votes
0answers
66 views

Solve $f(x)\mid g^2(x)+1$ in $\mathbb Z[x]$

We know that if $p\in \mathbb P$ and $p\equiv 1\bmod 4$ then we can find $t\in\mathbb Z$ such that $p\mid t^2+1.$ For what polynomial $f(x)\in \mathbb Z[x]$, we can find $g(x)\in \mathbb Z[x]$ ...
0
votes
1answer
62 views

work out the value of a - b from the identity $ax+18=2(x-b)$

How do I solve the following question? You are given the algebraic identity: $ax+18=2(x-b)$ Work out the values of $a-b$
2
votes
2answers
81 views

Ideals generated by roots of polynomials

Let $\alpha$ be a root of $x^3-2x+6$. Let $K=\mathbb{Q}[\alpha]$ and let denote by $\mathscr{O}_K$ the number ring of $K$. Now consider the ideal generated by $(4,\alpha^2,2\alpha,\alpha -3)$ in ...
3
votes
0answers
92 views

How does Dedekind's Theorem work for a prime dividing the discriminant of a number field?

Let $f \in \mathbb{Z}[x]$ be an irreducible monic polynomial, let $N$ be its splitting field, and let $G$ be the Galois group of the extension $N/\mathbb{Q}$. Let $p$ be a prime dividing the ...
3
votes
1answer
46 views

If $x^2+ax+b$ is an integer for every integer $x$ then comment on the coefficients $a$ and $b$ MCQ

Probably a more general category (number theory) multiple choice question but no clue how to get to a clear conclusion . Here's how it goes : Q) If $x^2+ax+b$ is an integer for every integer x ...
3
votes
2answers
100 views

Conjugates of $12^{1/5}+54^{1/5}-144^{1/5}+648^{1/5}$ over $\mathbb{Q}$

After much manual computation I found the minimal polynomial to be $x^5+330x-4170$, although I would very much like to know if there's a clever way to see this. I suspect there is from seeing that the ...
4
votes
1answer
35 views

How does the set of algebraic numbers compare to the set of possible fixed points for polynomials (with integer coefficients but not y=x)?

I was thinking of a way to map any polynomial $P$ with at least one real root onto some polynomial $Q$, s.t. the real roots of $P$ are exactly the real fixed points of $Q$, (There could be many, so we ...
2
votes
3answers
247 views

Way to show $x^n + y^n = z^n$ factorises as $(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n$

For odd $n$ the Fermat equation $x^n + y^n = z^n$ factorises as $$(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n,$$ where $\zeta = e^{2 \pi i/n}$. I tried seeing this was true by multiplying ...
0
votes
0answers
277 views

Polynomial Ring F[x] integrally closed?

Question: Let $F$ be a field and $A\subset F[x]$ the polynomials without the linear term. Prove that $F[x]$ is the integral closure of $A$. My Proof: Since we have $x=x^3/x^2$, the field of fractions ...
16
votes
3answers
282 views

How to prove summation, multiplication, subtraction of two roots of $1+x+\frac{x^2}{2!}+\cdots+\frac{x^p}{p!}=0$ aren't rationals?

Assume $a$, $b$ are distinct roots of the following equation: $$1+x+\frac{x^2}{2!}+\cdots+\frac{x^p}{p!}=0,$$ where $p$ is a prime number and $p \gt 2$. How to prove that $ab$, $a+b$, $a-b$ are not ...
3
votes
0answers
88 views

Given an integer, how can I detect the nearest integer perfect power efficiently?

If you give me an integer N, how can I detect the nearest integer perfect power, larger or smaller than N? In other words, the perfect power the distance between N and which is less than the ...
7
votes
1answer
119 views

“Real part” of a number field

Let ${\mathbb K} \subseteq {\mathbb C}$ be a finite extension of $\mathbb Q$, and let $n=[{\mathbb K} : {\mathbb Q}]$. Let $X_{\mathbb K}$ denote the set of all “components” (i.e., real and imaginary ...
2
votes
1answer
228 views

minimal polynomial of power of algebraic number

Consider any algebraic number $\alpha$ which is given by its minimal polynomial $f$. How can I compute the minimal polynomial of $\alpha^m$ for some natural number $m$? How efficient the algorithm is? ...
1
vote
0answers
311 views

Transforming root-equations into polynomials

Let's define special polynomials as polynomials in $\mathbb{Q}[X]$, where we allow to make roots, too. Examples: $\sqrt{X^4+1}$, $\sqrt[3]{X}+\sqrt{X+1}$, $\sqrt{X+\sqrt{X+1}}$ How can I transform a ...
3
votes
3answers
115 views

Proving $\frac{-\theta + \theta^2}{2}$ is an algebraic integer in $K = \mathbb{Q}(\theta)$, given that $\theta^3 + 11\theta - 4 = 0$

As the title says, given that $\theta^3 + 11\theta - 4 = 0$, I'm trying to prove that $\frac{-\theta + \theta^2}{2}$ is an algebraic integer in $K = \mathbb{Q}(\theta)$. I know that $x^3 + 11x -4$ ...
4
votes
6answers
296 views

How to prove that $\frac{10^{\frac{2}{3}}-1}{\sqrt{-3}}$ is an algebraic integer

As the title says, I'm trying to show that $\frac{10^{\frac{2}{3}}-1}{\sqrt{-3}}$ is an algebraic integer. I suppose there's probably some heavy duty classification theorems that give one line ...
4
votes
3answers
137 views

Showing that an algebraic number is not a root of a real

While answering this question on mathoverflow, I stumbled across a question that I expect may be easily answered by someone knowing a bit more algebra than me. Let's make it really specific. ...
3
votes
1answer
196 views

Polynomial interpolation of the residues of a rational function

Let $g(z) = a\prod_{i=1}^N (z-\lambda_i) \in \mathbb{Q}[z]$ be square-free. At each root $\lambda_i \in \mathbb{C}$, let $r_i$ denote the residue $\mathrm{Res}_{\lambda_i} 1/g(z)$. Let $I_g(z)$ ...
4
votes
1answer
374 views

rational angles with sines expressible with radicals

An angle x is rational when measured in degrees. sin(x) is can be written using radicals. What are the conditions on x? If nested square roots are allowed? What I know so far: If sin(x) can be ...
7
votes
3answers
533 views

Intuition regarding Chevalley-Warning Theorem

Three versions of the theorem are stated on pages 1-2 in these notes by Pete L. Clark: http://math.uga.edu/~pete/4400ChevalleyWarning.pdf Could anyone offer some intuitive way to think about this ...
3
votes
1answer
507 views

How many solutions are there to $F(n,m)=n^2+nm+m^2 = Q$?

Let $n,m$ be two positive integers, we consider: $$F(n,m)=n^2+nm+m^2$$ Let $Q$ be one value reach by $F(n,m)$. How many different pairs $(n,m)$ verify $F(n,m)=Q$?
3
votes
4answers
873 views

Simple formula for integer polynomial with $2\sin(2\pi/n)$ as a root?

Is there a simple formula an integer polynomial that $2\sin(2\pi/n)$ satisfies? For $2\cos(2\pi/n)$ the answer is relatively nice. For any given $n$, we have $2\cos(2\pi/n)= z + z^{-1}$ where $z = ...