Content deleted Content added
Line 188:
! colspan="9" | ...
|}
::When I wrote the section "The Proofs", my intent was to emphasize the proof of Cantor's second theorem, so I simplified his proof of the countability of algebraic numbers by leaving out "irreducible" so readers wouldn't have to know what an irreducible polynomial is. I'm sorry that you found my method less clear than Cantor's on the ordering of the algebraic numbers of a particular height. Using your enumeration table, the polynomials of height 2 give 0, -1, 1, 0 as roots, so the ordering will be -1, 0, 0, 1. In this enumeration, duplicates often appear within a height and between heights, but Cantor's proof of his second theorem does handle duplicates.
::However, you do bring up an excellent question: Shall we mention Cantor's use of irreducible polynomials? I see two ways to mention it: Add it to the text or add a footnote at the end of the paragraph that points out the text's ordering produces duplicates and that Cantor's original enumeration eliminates duplicates by using irreducible polynomials. By the way, the reason for some of the longer footnotes in this article was to explain points in more depth—readers just wanting the main points can skip the footnotes. Which is better in this case? I don't know. Maybe some readers can give us feedback.
::I like your enumeration table. A few suggestions: Label it "Cantor's enumeration of algebraic numbers". Change "not coprime" to "not irreducible". Coprime refers to a set of two or more integers so it doesn't apply to polynomials such as 2''x''. The definition of [[irreducible polynomial]] states that: "A polynomial with integer coefficients, or, more generally, with coefficients in a [[unique factorization ___domain]] ''F'' is said to be '''irreducible''' over ''F'' if it is not [[unit (ring theory)|invertible]] nor zero and cannot be factored into the product of two non-invertible polynomials with coefficients in ''F''." This definition factors: 2''x'' = (2)(''x'') and it factors: 2''x''+2 = (2)(''x''+1). Finally, the exponent 1 in your table always appears in gray and it's well understood that "''x''" means "''x''<sup>1</sup>". Try leaving out this exponent. I think this might visually simply your table. - [[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 01:58, 15 December 2013 (UTC)
| |||