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| $[ set.mm $] | ||
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| $( | ||
| #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# | ||
| Constructible numbers | ||
| #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# | ||
| This section defines the set of constructible points as complex numbers which | ||
| can be drawn starting from two points (we take ` 0 ` and ` 1 ` ), and taking | ||
| intersections of circles and lines. | ||
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| We initially define two sets: | ||
| <UL> | ||
| <LI>geometrically, by recursively adding the intersection points : this is | ||
| ` Constr ` .</LI> | ||
| <LI>algebraically, by using field extensions : this is ` ConstrAlg ` . </LI> | ||
| </UL> | ||
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| We then proceed by showing that those two sets are equal. | ||
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| This construction is useful for proving the impossibility of doubling the | ||
| cube ( ~ imp2cube ), and of angle trisection ( ~ imp3ang ) | ||
| $) | ||
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| $( All algebraic numbers admit a minimal polynomial. $) | ||
| $( minplyeu $p |- ( ph -> E! p e. ( Monic1p ` E ) A. q e. ( Monic1p ` E ) ( -> p = q ) ) $) | ||
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| $( Use infval $) | ||
| $( minplyf $) | ||
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| $( | ||
| -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- | ||
| Degree of a field extension | ||
| -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- | ||
| $) | ||
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| ${ | ||
| fldextdgirred.b $e |- B = ( Base ` F ) $. | ||
| fldextdgirred.0 $e |- .0. = ( 0g ` F ) $. | ||
| fldextdgirred.d $e |- D = ( deg1 ` F ) $. | ||
| fldextdgirred.o $e |- O = ( F evalSub1 E ) $. | ||
| fldextdgirred.m $e |- M = ( Poly1 ` F ) $. | ||
| fldextdgirred.k $e |- K = ( F |`s E ) $. | ||
| fldextdgirred.l $e |- L = ( F |`s ( F fldGen ( E u. { Z } ) ) ) $. | ||
| fldextdgirred.f $e |- ( ph -> F e. Field ) $. | ||
| fldextdgirred.e $e |- ( ph -> E e. ( SubDRing ` F ) ) $. | ||
| fldextdgirred.p $e |- ( ph -> P e. dom O ) $. | ||
| fldextdgirred.1 $e |- ( ph -> P e. ( Irred ` M ) ) $. | ||
| fldextdgirred.z $e |- ( ph -> Z e. ( B \ E ) ) $. | ||
| fldextdgirred.2 $e |- ( ph -> ( ( O ` P ) ` Z ) = .0. ) $. | ||
| $( If ` Z ` is the root of an irreducible polynomial ` P ` over a field | ||
| ` K ` , then the degree of the field extension ` [ K [ Z ] : K ] ` is | ||
| the degree of the polynomial ` P ` . $) | ||
| fldextdgirred $p |- ( ph -> ( L [:] K ) = ( D ` P ) ) $= ? $. | ||
| $} | ||
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| $( | ||
| -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- | ||
| Quadratic field extensions | ||
| -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- | ||
| $) | ||
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| $c /FldExt2 $. | ||
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| $( Extend class notation with the quadratic field extension relation. $) | ||
| cfldext2 $a class /FldExt2 $. | ||
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| ${ | ||
| $d e f $. | ||
| $( Define the quadratic field extension relation. Quadratic field | ||
| extensions are field extensions of degree 2. $) | ||
| df-fldext2 $a |- /FldExt2 = { <. e , f >. | ( e /FldExt f /\ ( e [:] f ) = 2 ) } $. | ||
| $} | ||
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| $( | ||
| -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- | ||
| Chain | ||
| -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- | ||
| $) | ||
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| $c Chain $. | ||
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| $( Extend class notation with the (finite) chain builder function. $) | ||
| cchn $a class Chain $. | ||
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| ${ | ||
| $d r c n $. | ||
| $( Define the (finite) chain builder function. A chain is defined to be a | ||
| sequence of objects, whereas each object is in a given relation with the | ||
| next one. $) | ||
| df-chn $a |- Chain = ( r e. _V |-> { c e. Word dom r | | ||
| A. n e. ( dom c \ { 0 } ) ( c ` ( n - 1 ) ) r ( c ` n ) } ) $. | ||
| $} | ||
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| $( | ||
| -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- | ||
| Constructible points | ||
| -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- | ||
| $) | ||
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| $c Constr $. | ||
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| $( Extend class notation with the set of geometrically constructible points. $) | ||
| cconstr $a class Constr $. | ||
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| $( Define the set of geometrically constructible points, by recursively adding | ||
| the line-line, line-circle and circle-circle intersections constructions | ||
| using points in a previous iteration. $) | ||
| df-constr $a |- Constr = ( rec ( ( s e. _V |-> { x e. CC | | ||
| ( E. a e. s E. b e. s E. c e. s E. d e. s E. t e. RR E. r e. RR ( x = ( a + ( t x. ( b - a ) ) ) /\ x = ( c + ( r x. ( d - c ) ) ) /\ -. ( Im ` ( ( * ` ( b - a ) ) x. ( d - c ) ) ) = 0 ) | ||
| \/ E. a e. s E. b e. s E. c e. s E. e e. s E. f e. s E. t e. RR ( x = ( a + ( t x. ( b - a ) ) ) /\ ( abs ` ( x - c ) ) = ( abs ` ( e - f ) ) ) | ||
| \/ E. a e. s E. b e. s E. c e. s E. d e. s E. e e. s E. f e. s E. t e. RR ( -. a = d /\ ( abs ` ( x - a ) ) = ( abs ` ( b - c ) ) /\ ( abs ` ( x - d ) ) = ( abs ` ( e - f ) ) ) ) | ||
| } ) , { 0 , 1 } ) " _om ) $. | ||
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| ${ | ||
| constrtow2.b $e |- B = ( Base ` F ) $. | ||
| constrtow2.q $e |- Q = ( CCfld |`s QQ ) $. | ||
| constrtow2.a $e |- ( ph -> A e. Constr ) $. | ||
| constrtow2.f $e |- ( ph -> F e. Field ) $. | ||
| constrtow2.1 $e |- ( ph -> A e. B ) $. | ||
| $( If an algebraically constructible point ` A ` is in a field ` F ` , then | ||
| there is a tower of quadratic field extensions from ` QQ ` to ` F ` . $) | ||
| constrtow2 $p |- ( ph -> E. t e. ( Chain ` `' /FldExt2 ) ( ( t ` 0 ) = Q | ||
| /\ ( lastS ` t ) = F ) ) $= ? $. | ||
| $} | ||
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| $( | ||
| -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- | ||
| Impossible constructions | ||
| -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- | ||
| $) | ||
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| $( Impossibility of doubling the cube. This is expressed by stating that the | ||
| cubic root of 2, which is the side length of a cube of volume ` 2 ` , is | ||
| not constructible. $) | ||
| imp2cube $p |- -. ( 2 ^c ( 1 / 3 ) ) e. Constr $= ? $. | ||
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| $( Impossibility of trisecting angles. This is expressed by stating that the | ||
| cosine of an angle of ` ( _pi / 9 ) ` which would be the third of the | ||
| constructible angle ` ( _pi / 3 ) ` , is not constructible. $) | ||
| imp3ang $p |- -. ( cos ` ( _pi / 9 ) ) e. Constr $= ? $. |
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