Using the First Isomorphism Theorem

[CurtBuck]

I'm trying to understand the first isomorphism theorem for groups.

Part of the examples given in the book is showing that Q[x]/(x^3-3) is isomorphic to {a+b*sqrt(3)}

As I understand it, by finding a homomorphism from Q[x] to {a+b*sqrt(3)} in which the kernel is x^3-3, the two are isomorphic.

I am struggling in finding the homomorphism from Q[x] to {a+b*sqrt(3)}

Any help would be great.

 

[Office_Shredder]

Are you sure it's not supposed to be Q[x]/(x²-3)?

 

[CurtBuck]

Yeah, it was supposed to be Q[x]/(x^2-3)

 

[Office_Shredder]

You need to find an element in Q[x]/(x²-3) whose square is equal to 3, i.e. you want to find some y such that y²+(x²-3) = 3+(x²-3) (y is a polynomial here). So there's really two steps here

1) Find y.
2) Use y to construct an isomorphism

 

[mathwonk]

the homomorphism sends x to sqrt(3).

when you mod out by f(x), you make f(x) = 0, so x becomes a root of f. since sqrt(3) is the root of x^2-3, x becomes sqrt(3).