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:::When I started this talk section, I had in mind the construction of rationals from integers, and I thought that algebraic numbers could be constructed from rationals in a similar way. The former is done by computing with pairs (''p'',''q'') ∈ ℤ×(ℤ\{0}) with the intended meaning ''p''/''q''; I thought the latter could be done by computing with polynomials, where one polynomial would denote one algebraic number, "viz. its root". Meanwhile I saw that even an irreducible polynomial has several roots, so that there can't be a one-to-one correspondence between polynomials and algebraic numbers, anyway. So I lost my original motivation for asking for irreducibility. Probably the proof is simplest in its current form; maybe a footenote could be added as you suggested.
:::In the enumeration table, I tried to distinguish several reasons for excluding a polynomial, a non-coprime set of coefficients being one of them, non-irreducibility being another one (admittely subsuming the former); when changing the table to produce duplicates these reasons would disappear, anyway. I used the gray parts to indicate (to myself, in the first place) the systematic way the polynomials are enumerated (nevertheless, I missed all polynomials containing ''x''<sup>3</sup> and ''x''<sup>4</sup>; see the new table; I hope it is complete now ...), but you are right: at least the exponent of "''x''<sup>1</sup>" isn't needed for that; I now deleted it. Concerning duplicates: should we have a reason "repetition" (or "duplicate"?) and not assign them a number; or should we assign them a number and mention somewhere that the enumeration is not bijective, but surjective, which suffices for countability? The former case would save some indentation space, since the ''x''<sup>4</sup> column could be immediately adjacent to the leftmost (number) column, as in each row at least one of them is empty. The latter case wouldn't save much, as "(-1 ± √5) / 2" (to be kept) is about as long as "repetition". - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 12:38, 16 December 2013 (UTC)
::I think some readers may find the current text ambiguous on the question of whether duplicates appear in the sequence (of course, it doesn't matter for applying his second theorem). There are two ways to eliminate duplicates and both give the same result. Below is my first attempt at a footnote to clarify the situation and to introduce readers to Cantor's approach and your table:
::"Using this ordering and placing only the first occurrence of an real algebraic number in the sequence produces a sequence without duplicates. Cantor obtained the same sequence by using [[irreducible polynomial]]s: INSERT YOUR TABLE HERE"
::Your table is looking better, some more suggestions: remove the "·" in 2·''x'', etc. In the enumeration, you can use ''x''<sub>1</sub> instead of "1.", etc. (This would connect your table closer to the article where all the sequences are ''x''<sub>1</sub>, ''x''<sub>2</sub>, ….) Also, in front of the first coefficient, you can leave out the "+" since every polynomial starts with a positive coefficient. Finally, concerning irreducible polynomials versus coprimes, I apologize for not being clearer. I should have quoted the following from "[[Irreducible polynomial]]":
::"It is helpful to compare irreducible polynomials to [[prime number]]s: prime numbers (together with the corresponding negative numbers of equal magnitude) are the irreducible [[integer]]s. They exhibit many of the general properties of the concept of 'irreducibility' that equally apply to irreducible polynomials, such as the essentially unique factorization into prime or irreducible factors:"
::This means that you factor 6''x'' = (2)(3)(''x''). Basically, the terms to use when working with factoring polynomials are "reducible" and "irreducible" (they are the counterparts to "composite" and "prime"). I think that you may be generalizing the term [[coprime]] to single integers to handle polynomials, such as 3''x'', when you call this polynomial "not coprime". I've done a Google search and I only found the term "coprime" referring two or more integers. So I think your table would be more accurate and clearer if you used the term "not irreducible". Also, I have the philosophy of placing minimal demands on the reader (whenever possible). By only using the word "irreducible", the reader is not required to understand "coprime".
::I hope you don't mind all my suggestions (I can be a bit of a perfectionist when it comes to tables). I think your table is an excellent addition to the article and will definitely help readers understand the ordering. In fact, it motivated me to reread Cantor's article and I noticed a detail that I had forgotten: Cantor gives the number of algebraic reals of heights 1, 2, and 3, which (of course) agree with your table. --[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 18:20, 17 December 2013 (UTC)
== Contrast 2nd theorem with sequence of rational numbers? ==
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