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

Revision as of 15:10, 13 December 2013

Revision as of 16:40, 13 December 2013

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[[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 15:10, 13 December 2013 (UTC)

:I made a list of algebraic numbers, ordered by Cantor's rank, as a collapsible table. Maybe it is illustrative to include it (in collapsed form) into the article. At least I my self learned (1) that "''irreducible''" should mean "''cannot be written as product of smaller polynomials'' '''with integer coefficients'''", and (2) an irreducible polynomial in that sense can well have several solutions; two facts that I should have remebered from my school time. - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 16:40, 13 December 2013 (UTC)

{| align="right" class="collapsible collapsed" style="text-align:left"

! colspan="9" | '''Enumeration of algebraic numbers'''

|-

! colspan="9" | '''Height 1:'''

|-

| '''1.''' ||           || || {{color|#a0a0a0|+1·}}''x''{{color|#a0a0a0|¹}} || {{color|#a0a0a0|+ 0}} || = 0 ||           || 0

|-

! colspan="9" | '''Height 2:'''

|-

|-

| || || || {{color|#c0c0c0|+}}2·''x''{{color|#c0c0c0|¹}} || {{color|#c0c0c0|+0}} || = 0 || || not coprime

|-

| '''2.''' || || || {{color|#c0c0c0|+1·}}''x''{{color|#c0c0c0|¹}} || +1 || = 0 || || -1

|-

| '''3.''' || || || {{color|#c0c0c0|+1·}}''x''{{color|#c0c0c0|¹}} || -1 || = 0 || || +1

|-

| || || {{color|#c0c0c0|+1·}}''x''² || {{color|#c0c0c0|+0·''x''¹}} || {{color|#c0c0c0|+ 0}} || = 0 || || not irreducible

|-

! colspan="9" | '''Height 3:'''

|-

| || || || {{color|#c0c0c0|+}}3·''x''{{color|#c0c0c0|¹}} || {{color|#c0c0c0|+0}} || = 0 || || not coprime

|-

| '''5.''' || || || {{color|#c0c0c0|+}}2·''x''{{color|#c0c0c0|¹}} || +1 || = 0 || || -1/2

|-

| '''6.''' || || || {{color|#c0c0c0|+}}2·''x''{{color|#c0c0c0|¹}} || -1 || = 0 || || +1/2

|-

| '''4.''' || || || {{color|#c0c0c0|+1·}}''x''{{color|#c0c0c0|¹}} || +2 || = 0 || || -2

|-

| '''7.''' || || || {{color|#c0c0c0|+1·}}''x''{{color|#c0c0c0|¹}} || -2 || = 0 || || +2

|-

| || || {{color|#c0c0c0|+}}2·''x''² || {{color|#c0c0c0|+0·''x''¹}} || {{color|#c0c0c0|+0}} || = 0 || || not coprime

|-

| || || {{color|#c0c0c0|+1·}}''x''² || +{{color|#c0c0c0|1·}}''x''{{color|#c0c0c0|¹}} || {{color|#c0c0c0|+0}} || = 0 || || not irreducible

|-

| || || {{color|#c0c0c0|+1·}}''x''² || -{{color|#c0c0c0|1·}}''x''{{color|#c0c0c0|¹}} || {{color|#c0c0c0|+0}} || = 0 || || not irreducible

|-

| || || {{color|#c0c0c0|+1·}}''x''² || {{color|#c0c0c0|+0·''x''¹}} || +1 || = 0 || || no real root

|-

| || || {{color|#c0c0c0|+1·}}''x''² || {{color|#c0c0c0|+0·''x''¹}} || -1 || = 0 || || not irreducible

|-

! colspan="9" | '''Height 4:'''

|-

| || || || {{color|#c0c0c0|+}}4·''x''{{color|#c0c0c0|¹}} || {{color|#c0c0c0|+0}} || = 0 || || not coprime

|-

| '''13.''' || || || {{color|#c0c0c0|+}}3·''x''{{color|#c0c0c0|¹}} || +1 || = 0 || || -1/3

|-

| '''14.''' || || || {{color|#c0c0c0|+}}3·''x''{{color|#c0c0c0|¹}} || -1 || = 0 || || +1/3

|-

| || || || {{color|#c0c0c0|+}}2·''x''{{color|#c0c0c0|¹}} || +2 || = 0 || || not coprime

|-

| || || || {{color|#c0c0c0|+}}2·''x''{{color|#c0c0c0|¹}} || -2 || = 0 || || not coprime

|-

| '''8.''' || || || {{color|#c0c0c0|+1·}}''x''{{color|#c0c0c0|¹}} || +3 || = 0 || || -3

|-

| '''19.''' || || || {{color|#c0c0c0|+1·}}''x''{{color|#c0c0c0|¹}} || -3 || = 0 || || +3

|-

| || || {{color|#c0c0c0|+}}3·''x''² || {{color|#c0c0c0|+0·''x''¹}} || {{color|#c0c0c0|+0}} || = 0 || || not coprime

|-

| || || {{color|#c0c0c0|+}}2·''x''² || +{{color|#c0c0c0|1·}}''x''{{color|#c0c0c0|¹}} || {{color|#c0c0c0|+0}} || = 0 || || not irreducible

|-

| || || {{color|#c0c0c0|+}}2·''x''² || |-{{color|#c0c0c0|1·}}''x''{{color|#c0c0c0|¹}} || {{color|#c0c0c0|+0}} || = 0 || || not irreducible

|-

| || || {{color|#c0c0c0|+}}2·''x''² || {{color|#c0c0c0|+0·''x''¹}} || +1 || = 0 || || no real root

|-

| '''11.,16.''' || || {{color|#c0c0c0|+}}2·''x''² || {{color|#c0c0c0|+0·''x''¹}} || -1 || = 0 || || ±1/√{{overline|2}}

|-

| || || {{color|#c0c0c0|+1·}}''x''² || +2·''x''{{color|#c0c0c0|¹}} || {{color|#c0c0c0|+0}} || = 0 || || not irreducible

|-

| || || {{color|#c0c0c0|+1·}}''x''² || -2·''x''{{color|#c0c0c0|¹}} || {{color|#c0c0c0|+0}} || = 0 || || not irreducible

|-

| || || {{color|#c0c0c0|+1·}}''x''² || +{{color|#c0c0c0|1·}}''x''{{color|#c0c0c0|¹}} || +1 || = 0 || || no real root

|-

| '''9.,15.''' || || {{color|#c0c0c0|+1·}}''x''² || +{{color|#c0c0c0|1·}}''x''{{color|#c0c0c0|¹}} || -1 || = 0 || || (-1 ± √{{overline|5}}) / 2

|-

| || || {{color|#c0c0c0|+1·}}''x''² || -{{color|#c0c0c0|1·}}''x''{{color|#c0c0c0|¹}} || +1 || = 0 || || no real root

|-

| '''12.,18.''' || || {{color|#c0c0c0|+1·}}''x''² || -{{color|#c0c0c0|1·}}''x''{{color|#c0c0c0|¹}} || -1 || = 0 || || (+1 ± √{{overline|5}}) / 2

|-

| || || {{color|#c0c0c0|+1·}}''x''² || {{color|#c0c0c0|+0·''x''¹}} || +2 || = 0 || || no real root

|-

| '''10.,17.''' || || {{color|#c0c0c0|+1·}}''x''² || {{color|#c0c0c0|+0·''x''¹}} || -2 || = 0 || || ±√{{overline|2}}

|-

! colspan="9" | '''Height 5:'''

|-

| || || || {{color|#c0c0c0|+}}5·''x''{{color|#c0c0c0|¹}} || {{color|#c0c0c0|+0}} || = 0 || || not coprime

|-

! colspan="9" | ...

|}