1. ## Algebraic - countable

A real number xR is called algebraic if there exists integers a0x^n+an1x^n1+.....+alx+a0=0.
Show that (2)^1/2,(2)^1/3, and 3+(2)^1/2 are algebraic.
Fix n in N and let An be the algebraic numbers obtained as roots of polynomials with integer coefficients that have degree n. Using the fact that every polynomial has a finite number of roots, show that is countable.
The algebraic definition is getting me confused.

2. Originally Posted by kathrynmath
A real number xR is called algebraic if there exists integers a0x^n+an1x^n1+.....+alx+a0=0.
Show that (2)^1/2,(2)^1/3, and 3+(2)^1/2 are algebraic.
Fix n in N and let An be the algebraic numbers obtained as roots of polynomials with integer coefficients that have degree n. Using the fact that every polynomial has a finite number of roots, show that is countable.
The algebraic definition is getting me confused.
The definition you quoted isn't formatted/typed correctly. You can look @ mathworld for one that's typeset properly

consider (x - sqrt(2)) * (x + sqrt(2)). Not sure how to handle 2^(1/3) and 3+2^(1/2) but someone else will probably jump on those.

For the second part see here

http://www.mathhelpforum.com/math-he...le-156204.html

Edit: I see from mathworld that the radical integers are a subring of the algebraic integers. If you wanted to prove that much stronger result then all the special cases asked for in the problem would immediately follow.

3. Originally Posted by kathrynmath
A real number xR is called algebraic if there exists integers a0x^n+an1x^n1+.....+alx+a0=0.
Show that (2)^1/2,(2)^1/3, and 3+(2)^1/2 are algebraic.
Fix n in N and let An be the algebraic numbers obtained as roots of polynomials with integer coefficients that have degree n. Using the fact that every polynomial has a finite number of roots, show that is countable.
The algebraic definition is getting me confused.

Put $x=2^{1/3}=\sqrt[3]{2}\Longrightarrow x^3=2\Longleftrightarrow x^3-2=0$ , and thus our element is a root of $f(t)=t^3-2$.

Try to do something similar with the other ones.

Tonio