Identifying type of field extension

[PsychonautQQ]

Homework Statement

Let [ S] = {2^(1/n) | for all n in the natural numbers}, is Q[ S] algebraic? finite? simple? separable?

Homework Equations

The Attempt at a Solution

I believe it is algebraic because every element of [ S] will be a root of x^n-2, and every element of Q is obviously algebraic over Q[X] and therefore Q[ S] will be algebraic.

I believe it is not finite because Q[ S] will be an infinite dimensional vector space over Q with basis {2^(1/2),2^(1/3),..., } up to infinity

I believe it is not simple because S is a whole set of linearly independent elements.

I'm not really sure about separable. I'm having a hard time with this partly because I can't think of a polynomial for which Q[ S] is the splitting field of, some polynomial of infinite degree surely?
But yeah, if anyone has any insight I'd appreciate it.

 

[fresh_42]

Homework Statement

Let [ S] = {2^(1/n) | for all n in the natural numbers}, is Q[ S] algebraic? finite? simple? separable?

Homework Equations

The Attempt at a Solution

I believe it is algebraic because every element of [ S] will be a root of x^n-2, and every element of Q is obviously algebraic over Q[X] and therefore Q[ S] will be algebraic.

I believe it is not finite because Q[ S] will be an infinite dimensional vector space over Q with basis {2^(1/2),2^(1/3),..., } up to infinity

I believe it is not simple because S is a whole set of linearly independent elements.

I'm not really sure about separable. I'm having a hard time with this partly because I can't think of a polynomial for which Q[ S] is the splitting field of, some polynomial of infinite degree surely?
But yeah, if anyone has any insight I'd appreciate it.

You already claimed that it is not finite. Can it be an extension of a single polynomial of finite degree? Polynomials of infinite degree don't exist and formal series are of no help here. However, this doesn't answer the question about separability. For this we need to know what a separable field extension is. Can you define it without referring to a single polynomial?

 

[andrewkirk]

Isn't there a theorem that a field $F$ can have an inseparable extension only if its characteristic is nonzero? If that's correct (I'm a bit sketchy on that part of Galois theory) then you can think about what the characteristic of $\mathbb Q$ is.

EDIT: I didn't see that Fresh had posted while I was mulling over this. Have a go at his suggestions first.