Algebraic geometry has many deep and powerful ideas. It started out by looking at the space described by the zeros of a polynomial, eg y^2 - x^3 - x = 0, x^2+y

Algebraic geometry has many deep and powerful ideas. It started out by looking at the space described by the zeros of a polynomial, eg y^2 - x^3 - x = 0, x^2+y^2+1=0 But actually you do not need to talk about the underlying space directly. If you want to talk about a space, all you actually need to think about are the possible functions on the space. If you want to talk about geometry, you only need the algebra of functions on that space, so in the example just the polynomials themselves, rather than having to say explicitly solve it for the points. You can use this big idea in a lot of other areas of mathematics and physics.