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:::I changed the table according to your suggestions (perfectionism in writing optimizes the overall workload, since the table is written only once, but read -hopefully- a lot of times). Maybe the indices like in ''x''<sub>'''3'''</sub> should not be in boldface? And: are you sure that no algebraic number may occur as root of two different irreducible polynomials? I've forgotten almost all my algebra knowledge... - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 20:40, 17 December 2013 (UTC)
::I like your attitude about perfectionism—I agree, we should think about the reader's workload. I also like the way you nicely simplified the table to have just 2 columns, by putting using "''x<sub>n</sub>'' =" with the roots. I think that ''x''<sub>3</sub> is preferable to ''x''<sub>'''3'''</sub> because the text doesn't use boldface and it looks better. Some other suggestions: I found double indexing "''x''<sub>11,16</sub>" confusing. Try "''x''<sub>11</sub>, ''x''<sub>16</sub>" or, perhaps better, "''x''<sub>16</sub>, ''x''<sub>11</sub>" to match the way that the + of the ± goes with ''x''<sub>16</sub>, and the – goes with ''x''<sub>11</sub> (or maybe there's a minus-plus symbol with minus on top of the plus). Also, I see no need for the large space between the "''x<sub>n</sub>'' =" and the roots at the top of the table. I can see you're lining up with the roots at the bottom of the table, but on a first reading, many users may not go to the bottom of the table and may wonder about the space. Finally, try moving the "…" over a bit at the end of the table.
::Your question about the possibility of an algebraic number occurring as the root of two different irreducible polynomials is very relevant. At the site: [http://www.encyclopediaofmath.org/index.php/Algebraic_number Algebraic Number (Encyclopedia of Math)], you can read about the minimal polynomial of an algebraic number. This minimal polynomial is the polynomial of least degree that has α as a root, has rational coefficients, and first coefficient 1. It is irreducible. By multiplying by the least common denominator of all its coefficients, you obtain α's irreducible polynomial with integer coefficients that Cantor uses. The minimal polynomial ''Φ(x)'' of the algebraic number α can be easily shown to be the factor of any polynomial ''p(x)'' with rational coefficients that has root α. You start by dividing ''p(x)'' by ''Φ(x)'' using long division. This gives: ''p(x)'' = ''q(x)'' ''Φ(x)'' + ''r(x)'' where deg(''r(x)'') < deg(''Φ(x)''). Assume ''r(x)'' ≠ 0. Since ''p''(α) = ''Φ''(α) = 0, we then have ''r''(α) = 0 which contradicts the fact that the minimal polynomial ''Φ(x)'' is the polynomial of least degree with root α. So ''r(x)'' must be 0. Therefore: ''p(x)'' = ''q(x)'' ''Φ(x)'' so the minimal polynomial is a factor of ''p(x)''. --[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 20:32, 18 December 2013 (UTC)
== Contrast 2nd theorem with sequence of rational numbers? ==
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