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I was reading Undergraduate Algebraic Geometry by Miles Reid.

and on page 1. It says

If $k$ is in $\textbf{R}$ or $\textbf{C}$ (which it quite often is).

Now I should state some previous information.

A variety is (roughly) a locus defined by polynomial equation: $$ V = \{P \in k^n | \text{f}_i(P) = 0\} \subset k^n $$ where $k$ is a field and $\text{f}_i \in k[X_1, \cdots , X_n] $ are polynomials; so for example the plane curves $\textbf{C} : (\text{f}(x,y) = 0) \subset \textbf{R}^2$ or $\textbf{C}^2$

Now the question:

When are polynomials not in either $\textbf{R} \mbox{ or } \textbf{C}$??

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Have you heard of fields of characteristic $p$? –  KReiser Sep 22 '12 at 14:01
    
@KReiser nope, not yet, just found the book at the bookstore and thought it would be a good to study. Got a good references or book? –  yiyi Sep 22 '12 at 14:03
    
Well, your polynomial could be defined over a field of characteristic $p$, which does not embed into either $\mathbb{R}$ or $\mathbb{C}$. For example, consider $x^2-x-1$ over $\mathbb{F}_3$, the field with 3 elements. –  KReiser Sep 22 '12 at 14:06

1 Answer 1

up vote 5 down vote accepted

I think the point is that the definition of a variety makes sense for any (algebraically closed) field $k$. But for most concrete examples that you will encounter in the class the field will be the real or the complex numbers.

Edit: To add a bit more on "how can a polynomial not be in the reals or the complex" numbers?

There are a lot of other fields than the real and the complex numbers. You can take for example the rational numbers. They also constitute a field.

Consider the finite field $\mathbb{Z} / 3\mathbb{Z}$. This field has three elements: $[0], [1], [2]$ and we have for example that $[2][2] = [4] = [1]$. You can consider polynomials over this field, for example: $$ f(x) = [1]X^2 + [2]X.$$ (we will usually write this just as $f(X) = X^2 + 2X$). The polynomial is not "in the reals or the complex" because the coefficients are not from those fields.

Wikipedia has a page on field with a list of example of different fields.

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So which fields are not in the Reals or Complex, could you give me a name to look up? –  yiyi Sep 22 '12 at 14:07
    
@MaoYiyi: Did my edit help? –  Thomas Sep 22 '12 at 14:10
    
Oh, I get it know. I was confused because I was thinking along the lines of $x=$ (anything not int he reals) but $x^2+2x$ is in either the Reals or Complex, so how can any polynomial not be in the Reals or Complex. –  yiyi Sep 22 '12 at 14:15
    
@MaoYiyi As listed, it is not true that $x^2+2x$ with coefficients from $\mathbb{F}_3$ is a polynomial with real or complex coefficients, because the coefficients are from $\mathbb{F}_3$, which does not embed into the real or the complex numbers. –  KReiser Sep 22 '12 at 15:57

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