Talk:Algebraic number

Subfields

@Nxavar: Note our definition of subfield:

Let L be a field. A subfield of L is a subset K of L that is closed under the field operations of L and under taking inverses in L. In other words, K is a field with respect to the field operations inherited from L.

The algebraic numbers are a subfield of the complex numbers; we need not verify the field axioms. — Arthur Rubin (talk) 16:49, 27 January 2016 (UTC)

The problem is that it is not mentioned that the complex numbers are a field, which makes the argument incomplete. Nxavar (talk) 08:40, 28 January 2016 (UTC)

Assessment comment (2006)

The comment(s) below were originally left at Talk:Algebraic number/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

The introduction defines algebraic numbers as being over the rational numbers, then says that they can be over other fields. Lacks discussion of very rich history and motivation, this article will be a little confusing to anyone that is not a mathematician shotwell 16:23, 7 October 2006 (UTC)

Last edited at 22:13, 19 April 2007 (UTC). Substituted at 01:45, 5 May 2016 (UTC)

Real and Irrational

The are some disagreement between these two statements in wiki pages:

algebraic numbers: "A real algebraic number of degree 2 is a quadratic irrational."

quadratic irrational: "When c is positive, we get real quadratic irrational numbers, while a negative c gives complex quadratic irrational numbers which are not real numbers."

I suggest, to skip confusion, modify this page to "An algebraic number of degree 2 is a quadratic irrational." — Preceding unsigned comment added by 88.3.70.40 (talk) 09:29, 1 January 2017 (UTC)

Pi as an example

Recently someone put pi as an example of algebraic number, either because he/she did not understand the reference or a grief. Since it causes direct contradiction, I removed it for now. --Raxu360 (talk) 00:30, 30 June 2019 (UTC)

The malaise of Wikipedia mathematics articles.

An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently, by clearing denominators, with integer coefficients).

As with so many mathematical definitions in Wikipedia, this definition can only be understood by those who already know what the term means, and likely not all of them. (I have seen mathematical definitions in Wikipedia where I already knew what the term meant, but still couldn't understand the definition.)

The sole purpose of an encyclopedia is to convey information to those who do not already have it. This definition does not perform that function. Koro Neil (talk) 21:06, 8 October 2021 (UTC)

I would include that it is not the same as number representable in radicals. Valery Zapolodov (talk) 05:56, 25 December 2022 (UTC)

Do you have a proposed alternative? My suggestion would be to just get rid of all of the parentheticals as they make the sentence very awkward. Something like: “An algebraic number is a complex number that is the solution to a single-variable polynomial equation with rational coefficients. [... examples and further explanation unpacking some of those terms ...]” It looks like the definition has changed somewhat since this comment though. Did those changes solve your issue? –jacobolus (t) 00:43, 26 December 2022 (UTC)

Can we link to an online proof that they are enumerable

Says "The set of algebraic numbers is countable (enumerable)," with two old print refs. Can we link to an online proof ? - Rod57 (talk) 22:40, 19 September 2022 (UTC)

What online proof? Math. Stack. is not allowed, cause it is not a reliable source. Valery Zapolodov (talk) 05:54, 25 December 2022 (UTC)