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)

Abel, Galois, Liouville

This article begs for a note about the root operation + the four arithmetic operations, about the algebraic numbers of degree less than 5. One could also mention the Liouville's theorem about the poor approximation of algebraic numbers by rational numbers. Wlodzimierz Holsztynski 03:34, 27 December 2007 (CST)

Advanced Subpage

Here is another example where I think we could do with an advanced subpage. I think vector space proofs that the algebraic numbers form a field would be more appropriate in this context. Does anyone agree?Barry R. Smith 12:50, 8 May 2008 (CDT)

Terminology

I chose the words "defining polynomial" rather than "minimal polynomial" as the first term. Googling them both suggests minimal polynomial is more prevalent. However, this might not be the best criterion for making a decision. Both terms are somewhat descriptive, in different ways. Any preference? (Perhaps "defining polynomial" is better, because it won't need disambiguation with the use of "minimal polynomial" coming from linear algebra?)

Also, googling "linear space" and "vector space" suggests that "vector space" is more prevalent. Should we leave the advanced material in terms of linear spaces, or switch to the more common term?