Talk:Cantor's first set theory article/Archive 1: Difference between revisions

Revision as of 00:30, 19 December 2013

Revision as of 14:10, 27 December 2013

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! colspan="10" | '''Height 1:'''

|-

| || || || align="right" | {{color|#c0c0c0|1}}''x'' || {{color|#c0c0c0|+0}} || = 0 ||           || ''x''_'''1''' || = 0

|-

! colspan="10" | '''Height 2:'''

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| || || || align="right" | 2''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || || align="right" | {{color|#c0c0c0|1}}''x'' || +1 || = 0 || || ''x''_'''2''' || = −1

|-

| || || || align="right" | {{color|#c0c0c0|1}}''x'' || −1 || = 0 || || ''x''_'''3''' || = +1

|-

||| || align="right" | {{color|#c0c0c0|1}}''x''² || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

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||| || || align="right" | 3''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || || align="right" | 2''x'' || +1 || = 0 || || ''x''_'''5''' || = −1/2

|-

| || || || align="right" | 2''x'' || −1 || = 0 || || ''x''_'''6''' || = +1/2

|-

| || || || align="right" | {{color|#c0c0c0|1}}''x'' || +2 || = 0 || || ''x''_'''4''' || = −2

|-

| || || || align="right" | {{color|#c0c0c0|1}}''x'' || −2 || = 0 || || ''x''_'''7''' || = +2

|-

| || || align="right" | 2''x''² || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

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| || || || align="right" | 4''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || || align="right" | 3''x'' || +1 || = 0 || || ''x''_'''13''' || = −1/3

|-

| || || || align="right" | 3''x'' || −1 || = 0 || || ''x''_'''14''' || = +1/3

|-

| || || || align="right" | 2''x'' || +2 || = 0 || || colspan="2" | reducible

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| || || || align="right" | 2''x'' || −2 || = 0 || || colspan="2" | reducible

|-

| || || || align="right" | {{color|#c0c0c0|1}}''x'' || +3 || = 0 || || ''x''_'''8''' || = −3

|-

| || || || align="right" | {{color|#c0c0c0|1}}''x'' || −3 || = 0 || || ''x''_'''19''' || = +3

|-

| || || align="right" | 3''x''² || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

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| || || align="right" | 2''x''² || {{color|#c0c0c0|+0''x''}} || +1 || = 0 || || colspan="2" | no real root

|-

| || || align="right" | 2''x''² || {{color|#c0c0c0|+0''x''}} || −1 || = 0 || || ''x''_'''11'''16, '''16'x''₁₁ || = ±1/√{{overline|2}}

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || +2''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

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| || || align="right" | {{color|#c0c0c0|1}}''x''² || +{{color|#c0c0c0|1}}''x'' || +1 || = 0 || || colspan="2" | no real root

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || +{{color|#c0c0c0|1}}''x'' || −1 || = 0 || || ''x''_'''9'''15, '''15'x''₉ || = (−1 ± √{{overline|5}}) / 2

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || −{{color|#c0c0c0|1}}''x'' || +1 || = 0 || || colspan="2" | no real root

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || −{{color|#c0c0c0|1}}''x'' || −1 || = 0 || || ''x''_'''12'''18, '''18'x''₁₂ || = (+1 ± √{{overline|5}}) / 2

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || {{color|#c0c0c0|+0''x''}} || +2 || = 0 || || colspan="2" | no real root

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || {{color|#c0c0c0|+0''x''}} || −2 || = 0 || || ''x''_'''10'''17, '''17'x''₁₀ || = ±√{{overline|2}}

|-

| || align="right" | 2''x''³ || {{color|#c0c0c0|+0''x''²}} || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

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| || || || align="right" | 5''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || || || || ''':'''

! colspan="10" | ...

|}

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::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 a 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)

:::I didn't have web access during xmas holidays, but now I updated the table according to your recent suggestions. There is a "∓" symbol, but I think it looks unusual in an expression, so I instead changed the order of the lhs variables. I moved the final dots into the "=" column and simulated vertical dots by a colon, as I couldn't find an appropriate symbol or template.

:::I like your suggestion for a footnote containing our table. As you are currently editing the article anyway, would you insert your footnote and move the table? Maybe it is best to remove it from the talk page, to avoid confusion about where to do possible later table edits.

:::Last not least: Thank you for your explanation why there is only one minimal irreducible polynomial for an algebraic number; it helped me to bring back my memories about algebra. - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 14:09, 27 December 2013 (UTC)

== Contrast 2nd theorem with sequence of rational numbers? ==