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
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 , we say the characteristic is 0. A field of positive characteristic need not be finite. Two examples are the algebraic closure of (usually written when the emphasis is on being a field). Another basic example is the field of rational functions in one variable, . 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 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)
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)
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?Barry R. Smith 14:01, 10 May 2008 (CDT)
Algebraic numbers Complex numbers?
The article was edited a while back, changing the statement "algebraic numbers can be complex" to the statement "an algebraic number is a complex number...", with the rationale that the second was correct and stronger.
However, it is not quite correct. An algebraic number can always be viewed as a number in some algebraically closure of the rational numbers, and all of these closures are isomorphic fields. However, there is no canonical description of such a closure. Any such closure can be viewed as embedded in the complex numbers, but again, there is no canonical way to view this embedding. Thus, an algebraic number can generally be thought of as a complex number, but not in a canonical way, and in general it is just an element of some algebraic closure of the rationals.
Since all algebraic closures are isomorphic, this may seem like a trivial point, but it's not. It plays a serious role in trying to write computational number theory programs. For instance, viewing the rational numbers as embedded in the complexes in the usual way, the roots of can be thought of as the real number and the two non-real roots , where is a cube root of unity. Here, one intuitively sees a huge distinction between the three roots -- the first is real, and the second two aren't. However, this distinction is an artifact of having fixed the complex numbers as an algebraically closed field containing the rationals, and having an intuitive feel for the distinction between real and non-real numbers.
To work with these numbers algebraically in a computer, you cannot ahead of time "explain" the intuitive notion of the complex numbers to the computer ahead of time, and then allow it to work with algebraic numbers of arbitrarily large degree by working within this field (at least, as far as I know no one can do this). Instead, all you tell the computer is that you are working with a root of . Not only does the computer have no way to distinguish between the three roots from this description, there is no way for the computer to single out an algebraically closed field containing these roots.
Avoiding thinking of algebraic numbers as living in the complex numbers with a particular embedding is also important for theoretical reasons. Doing this is sort of like "fixing a basis" for a vector space when you don't have to -- working in a coordinate-free manner can often be important.
The above discussion is rather technical, from the point of view of someone who doesn't know much or anything about algebraic numbers and wants to learn from this page. To avoid much confusion, especially in the first sentence of the page, how about just referring to the algebraic number as a root of a polynomial. Discussion of what field to view this number in can be deferred until later in the article, at least not in the introduction. Any thoughts?Barry R. Smith 18:06, 7 December 2008 (UTC)
- I created an advanced subpage to deal with this discrepancy. Two things seem clear to me, from looking at several number theory books: 1. Many authors still define the algebraic numbers to be a subset of the complex numbers, and of course, this was how they were considered for a rather large chunk of history; 2. The best definition that avoids this assumption, and the one used by virtually all other authors, defines an algebraic number to be a number in a finite extension field of the rationals. Because "field" is advanced for a page aimed at undergraduate level people (they should be able to understand "algebraic number", even if "algebraic number theory" gets rather too advanced for this level), I decided to put this definition on an advanced subpage. So I rescind my objection to requiring algebraic numbers to be complex on the main page.