An injective function going from N to the set of algebraic numbers

[hangainlover]

Homework Statement

Prove that the set of algebraic numbers is countably infinite.

Homework Equations

If there exists a bijective map between N and a set A, N and A have the same cardinality

The Attempt at a Solution

Rather than coming up with a bijective map between S =the set of algebraic numers and N =natural numbers, I proved S is countable but i also have to prove that S is infinite.
So, I wanted to design an injective function f:N to S.
Can anyone come up with sucn an injective function?

 

[morphism]

How about the map that sends n to n...?

 

[hangainlover]

Ahh...
$f(x,n)=x-n$\\
$k(n) ={a:f(a,n)=0}$\\

Then, $k(1)=1, k(2)=2,k(3)=3, ... k(n) = n$.\\
this should be an injective map from N to the set of algebraic numbers...

 

[hangainlover]

thank you so much!

 

[morphism]

No problem.

But really you shouldn't be thinking of maps and such: just try to think about why there are infinitely many algebraic numbers. Every rational number is algebraic. But there's more. Every number of the form $\sqrt[n]{r}$ with r rational is also algebraic. And there's more still..