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### 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?
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### 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/...
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### 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 ...
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 $x,y \in \mathbb R$?
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### 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 ...
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 ...
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 ...
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### 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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### 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.
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 ...