Talk:Algebraic number

things to add:

• links to "rational number" and "polynomial"
• a couple of examples - and put in the polynomial that sqrt(2) satisfies
• mention that some, but not all, algebraic numbers can be expressed using radicals - mention and link Galois
• the link to "countable" should probably point to a new page on cardinality

I think the link should be to Countable set. Andres Luure 03:09, 26 March 2007 (CDT)

characteristic?

In this sentence: "The algebraic numbers form a field; in fact, they are the smallest algebraically closed field with characteristic 0. " I don't know what "characteristic 0" means. Perhaps a definition or a link would be helpful. --Catherine Woodgold 21:23, 28 April 2007 (CDT)

Oh, sorry. if 1 + 1 = 0 the characteistic is said to be 2, if 1 + 1 + 1 = 0 the characteristic is said to be 3, and forth. If there is no n such that $\scriptstyle n\cdot 1=0$, we say the characteristic is 0. A field of positive characteristic need not be finite. Two examples are the algebraic closure of $\mathbb {Z} _{2}$ (usually written $\mathbb {F} _{2}$ when the emphasis is on being a field). Another basic example is the field of rational functions in one variable, $\mathbb {F} _{2}(x)$. Fields of positive characteristic are important in applications to number theory. Greg Woodhouse 22:04, 28 April 2007 (CDT)

Thanks. I added part of your explanation as a footnote. I suppose $\mathbb {Z} _{2}$ means the integers modulo 2, a field of 2 elements. Hard to imagine the algebraic closure of that. --Catherine Woodgold 11:11, 29 April 2007 (CDT)

Yeah, it's an infinite extension. Greg Woodhouse 11:20, 29 April 2007 (CDT)