Linked Questions

0
votes
1answer
36 views

Product of Algebraic Number and Rational Number

If $a$ is an algebraic number and $b$ is a rational number, then show that $ab$ is algebraic number. My attempt is to prove it by contradiction, but it failed. Can someone please help me?
2
votes
1answer
68 views

How to prove that the sum and product of algebraic integers is an algebraic integer? [duplicate]

I would like to understand why the sum and product of algebraic integers are algebraic integers. For algebraic numbers (not integers) there is the wonderful website https://www.dpmms.cam.ac.uk/~wtg10/...
0
votes
1answer
28 views

Sums of conjugates of algebraic numbers

I would like to know if there is an elementary proof (without Galois theory, i.e. using the fact that conjugates are images by the base field automorphisms) of the fact that the conjugates of a sum of ...
1
vote
0answers
30 views

Algebraic numbers theory [duplicate]

How I can prove that if there were $x$ and $y$ are algebraic numbers then $x+y$ and $x \cdot y$ are also algebraic numbers given that $x,y \in \mathbb R$?
0
votes
0answers
64 views

Sum & Product of Two Algebraic Numbers [duplicate]

If you have algebraic numbers $x$ and $y$, and you know the polynomials of least degree of which each is a solution (written as a vectors of coefficients), then how is the vector of coefficients of ...
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 ...
2
votes
1answer
146 views

If $\alpha$ and $\beta$ are roots of monic polynomials in $\mathbb{Z}[x]$, is $\alpha + \beta$? [duplicate]

If $\alpha$ and $\beta$ are roots of monic polynomials (not necessarily the same polynomial) in $\mathbb{Z}[x]$, is $\alpha + \beta$? I know that $\alpha + \beta$ will be the root of some monic ...
3
votes
3answers
86 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, ...
1
vote
2answers
196 views

Addition of two algebraic integer numbers [duplicate]

Is the addition of two algebraic integer numbers also algebraic? I, guess it is, but i can't prove it. I wonder if multiplication of them is also algebraic.
0
votes
0answers
544 views

Set of algebraic integer form a ring.

An algebraic integer is a complex number that is the root of monic polynomial with integer coefficients. Show that the set of algebraic integers is a subring of $C$. (Hint: Use symmetric function ...
4
votes
3answers
124 views

construct polynomial from other polynomials

If I have a polynomial, P, with root $a$ and a polynomial, Q, with root $b$, is there a way to construct polynomial R such that $a+b$ is a root of R? Here's a concrete example. a = $\sqrt2$. $P(x) = ...
2
votes
4answers
173 views

Prove that $\sqrt[3]{2} +\sqrt{5}$ is algebraic over $\mathbb Q$

Prove that $\sqrt[3]{2} +\sqrt{5}$ is algebraic over $\mathbb{Q}$ (by finding a nonzero polynomial $p(x)$ with coefficients in $\mathbb{Q}$ which has $\sqrt[3] 2+\sqrt 5$ as a root). I first tried ...
15
votes
2answers
797 views

Decomposition of an algebraic number into a sum or product of algebraic numbers with smaller degree

An algebraic number can be identified by its minimal polynomial together with isolating intervals with rational bounds for its real and imaginary parts. The degree of an algebraic number is the degree ...
1
vote
0answers
40 views

Integral elements form a ring. What can we say about polynomials of sum and product? [duplicate]

Let $B$ be a ring (commutative and with identity). It is a standard fact in Algebraic Number Theory that the sum $b_{1}+b_{2}$ and the product $b_{1}b_{2}$ of integral elements $b_{1},b_{2}\in B$ over ...
10
votes
3answers
1k views

How to prove that algebraic numbers form a field?

I'd like to know how to prove algebraic numbers form a field, i.e., if $a,b$ are algebraic numbers, prove that $ab$ and $a+b$ are also algebraic numbers by finding specific polynomials that ...

15 30 50 per page