:: Primitive Roots of Unity and Cyclotomic Polynomials
:: by Broderick Arneson and Piotr Rudnicki
::
:: Received December 30, 2003
:: Copyright (c) 2003-2021 Association of Mizar Users

Lm1: for k being Element of NAT holds
( not k is zero iff 1 <= k )

proof end;

scheme :: UNIROOTS:sch 1
CompIndNE{ P₁[ non zero Nat] } :

for k being non zero Element of NAT holds P₁[k]

provided

A1: for k being non zero Element of NAT st ( for n being non zero Element of NAT st n < k holds
P₁[n] ) holds
P₁[k]

proof end;

theorem Th1: :: UNIROOTS:1

for f being FinSequence st 1 <= len f holds
f | (Seg 1) = <*(f . 1)*>

proof end;

Lm2: for f being FinSequence
for n, i being Element of NAT st i <= n holds
(f | (Seg n)) | (Seg i) = f | (Seg i)
by FINSEQ_1:5, RELAT_1:74;

theorem Th2: :: UNIROOTS:2

for f being FinSequence of F_Complex
for g being FinSequence of REAL st len f = len g & ( for i being Element of NAT st i in dom f holds
|.(f /. i).| = g . i ) holds
|.(Product f).| = Product g

proof end;

theorem Th3: :: UNIROOTS:3

for s being non empty finite Subset of F_Complex
for x being Element of F_Complex
for r being FinSequence of REAL st len r = card s & ( for i being Element of NAT
for c being Element of F_Complex st i in dom r & c = (canFS s) . i holds
r . i = |.(x - c).| ) holds
|.(eval ((poly_with_roots ((s,1) -bag)),x)).| = Product r

proof end;

theorem Th4: :: UNIROOTS:4

for f being FinSequence of F_Complex st ( for i being Element of NAT st i in dom f holds
f . i is integer ) holds
Sum f is integer

proof end;

theorem :: UNIROOTS:5

for x, y being Element of F_Complex
for r1, r2 being Real st r1 = x & r2 = y holds
( r1 * r2 = x * y & r1 + r2 = x + y ) ;

theorem Th6: :: UNIROOTS:6

for q being Real st q > 0 holds
for r being Element of F_Complex st |.r.| = 1 & r <> [**1,0**] holds
|.([**q,0**] - r).| > q - 1

proof end;

theorem Th7: :: UNIROOTS:7

for ps being non empty FinSequence of REAL
for x being Real st x >= 1 & ( for i being Element of NAT st i in dom ps holds
ps . i > x ) holds
Product ps > x

proof end;

theorem Th8: :: UNIROOTS:8

for n being Element of NAT holds 1_ F_Complex = (power F_Complex) . ((1_ F_Complex),n)

proof end;

theorem Th9: :: UNIROOTS:9

for n, i being Nat holds
( cos (((2 * PI) * i) / n) = cos (((2 * PI) * (i mod n)) / n) & sin (((2 * PI) * i) / n) = sin (((2 * PI) * (i mod n)) / n) )

proof end;

theorem Th10: :: UNIROOTS:10

for n, i being Nat holds [**(cos (((2 * PI) * i) / n)),(sin (((2 * PI) * i) / n))**] = [**(cos (((2 * PI) * (i mod n)) / n)),(sin (((2 * PI) * (i mod n)) / n))**]

proof end;

theorem Th11: :: UNIROOTS:11

for n, i, j being Element of NAT holds [**(cos (((2 * PI) * i) / n)),(sin (((2 * PI) * i) / n))**] * [**(cos (((2 * PI) * j) / n)),(sin (((2 * PI) * j) / n))**] = [**(cos (((2 * PI) * ((i + j) mod n)) / n)),(sin (((2 * PI) * ((i + j) mod n)) / n))**]

proof end;

theorem Th12: :: UNIROOTS:12

for L being non empty unital associative multMagma
for x being Element of L
for n, m being Nat holds (power L) . (x,(n * m)) = (power L) . (((power L) . (x,n)),m)

proof end;

theorem Th13: :: UNIROOTS:13

for n being Element of NAT
for x being Element of F_Complex st x is Integer holds
(power F_Complex) . (x,n) is Integer

proof end;

theorem Th14: :: UNIROOTS:14

for F being FinSequence of F_Complex st ( for i being Element of NAT st i in dom F holds
F . i is Integer ) holds
Sum F is Integer

proof end;

Lm3: Z_3 is finite
by MOD_2:def 20;

registration

cluster non empty non degenerated non trivial finite right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital V181() left_unital V191() V192() V193() domRing-like for doubleLoopStr ;

existence
ex b₁ being Field st b₁ is finite by Lm3, MOD_2:27;

cluster non empty non degenerated non trivial finite right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital V181() left_unital V191() V192() V193() for doubleLoopStr ;

existence
ex b₁ being Skew-Field st b₁ is finite by Lm3, MOD_2:27;

end;

definition

let R be Skew-Field;

func MultGroup R -> strict Group means :Def1: :: UNIROOTS:def 1
( the carrier of it = NonZero R & the multF of it = the multF of R || the carrier of it );

existence
ex b₁ being strict Group st
( the carrier of b₁ = NonZero R & the multF of b₁ = the multF of R || the carrier of b₁ )

proof end;

uniqueness
for b₁, b₂ being strict Group st the carrier of b₁ = NonZero R & the multF of b₁ = the multF of R || the carrier of b₁ & the carrier of b₂ = NonZero R & the multF of b₂ = the multF of R || the carrier of b₂ holds
b₁ = b₂ ;

end;

:: deftheorem Def1 defines MultGroup UNIROOTS:def 1 :
for R being Skew-Field
for b₂ being strict Group holds
( b₂ = MultGroup R iff ( the carrier of b₂ = NonZero R & the multF of b₂ = the multF of R || the carrier of b₂ ) );

theorem :: UNIROOTS:15

for R being Skew-Field holds the carrier of R = the carrier of (MultGroup R) \/ {(0. R)}

proof end;

theorem Th16: :: UNIROOTS:16

for R being Skew-Field
for a, b being Element of R
for c, d being Element of (MultGroup R) st a = c & b = d holds
c * d = a * b

proof end;

theorem Th17: :: UNIROOTS:17

for R being Skew-Field holds 1_ R = 1_ (MultGroup R)

proof end;

registration

let R be finite Skew-Field;

cluster MultGroup R -> finite strict ;

correctness
coherence
MultGroup R is finite ;

proof end;

end;

theorem :: UNIROOTS:18

for R being finite Skew-Field holds card (MultGroup R) = (card R) - 1

proof end;

theorem Th19: :: UNIROOTS:19

for R being Skew-Field
for s being set st s in the carrier of (MultGroup R) holds
s in the carrier of R

proof end;

theorem :: UNIROOTS:20

for R being Skew-Field holds the carrier of (MultGroup R) c= the carrier of R

proof end;

definition

let n be non zero Element of NAT ;

func n -roots_of_1 -> Subset of F_Complex equals :: UNIROOTS:def 2
{ x where x is Element of F_Complex : x is CRoot of n, 1_ F_Complex } ;

coherence
{ x where x is Element of F_Complex : x is CRoot of n, 1_ F_Complex } is Subset of F_Complex

proof end;

end;

:: deftheorem defines -roots_of_1 UNIROOTS:def 2 :
for n being non zero Element of NAT holds n -roots_of_1 = { x where x is Element of F_Complex : x is CRoot of n, 1_ F_Complex } ;

theorem Th21: :: UNIROOTS:21

for n being non zero Element of NAT
for x being Element of F_Complex holds
( x in n -roots_of_1 iff x is CRoot of n, 1_ F_Complex )

proof end;

theorem Th22: :: UNIROOTS:22

for n being non zero Element of NAT holds 1_ F_Complex in n -roots_of_1

proof end;

theorem Th23: :: UNIROOTS:23

for n being non zero Element of NAT
for x being Element of F_Complex st x in n -roots_of_1 holds
|.x.| = 1

proof end;

theorem Th24: :: UNIROOTS:24

for n being non zero Element of NAT
for x being Element of F_Complex holds
( x in n -roots_of_1 iff ex k being Element of NAT st x = [**(cos (((2 * PI) * k) / n)),(sin (((2 * PI) * k) / n))**] )

proof end;

theorem Th25: :: UNIROOTS:25

for n being non zero Element of NAT
for x, y being Element of COMPLEX st x in n -roots_of_1 & y in n -roots_of_1 holds
x * y in n -roots_of_1

proof end;

theorem Th26: :: UNIROOTS:26

for n being non zero Element of NAT holds n -roots_of_1 = { [**(cos (((2 * PI) * k) / n)),(sin (((2 * PI) * k) / n))**] where k is Element of NAT : k < n }

proof end;

theorem Th27: :: UNIROOTS:27

for n being non zero Element of NAT holds card (n -roots_of_1) = n

proof end;

registration

let n be non zero Element of NAT ;

cluster n -roots_of_1 -> non empty ;

correctness
coherence
not n -roots_of_1 is empty ;
by Th22;

cluster n -roots_of_1 -> finite ;

correctness
coherence
n -roots_of_1 is finite ;

proof end;

end;

theorem Th28: :: UNIROOTS:28

for n, ni being non zero Element of NAT st ni divides n holds
ni -roots_of_1 c= n -roots_of_1

proof end;

theorem Th29: :: UNIROOTS:29

for R being Skew-Field
for x being Element of (MultGroup R)
for y being Element of R st y = x holds
for k being Nat holds (power (MultGroup R)) . (x,k) = (power R) . (y,k)

proof end;

theorem Th30: :: UNIROOTS:30

for n being non zero Element of NAT
for x being Element of (MultGroup F_Complex) st x in n -roots_of_1 holds
not x is being_of_order_0

proof end;

theorem Th31: :: UNIROOTS:31

for n being non zero Element of NAT
for k being Element of NAT
for x being Element of (MultGroup F_Complex) st x = [**(cos (((2 * PI) * k) / n)),(sin (((2 * PI) * k) / n))**] holds
ord x = n div (k gcd n)

proof end;

theorem Th32: :: UNIROOTS:32

for n being non zero Element of NAT holds n -roots_of_1 c= the carrier of (MultGroup F_Complex)

proof end;

theorem :: UNIROOTS:33

for n being non zero Element of NAT ex x being Element of (MultGroup F_Complex) st ord x = n

proof end;

theorem Th34: :: UNIROOTS:34

for n being non zero Element of NAT
for x being Element of (MultGroup F_Complex) holds
( ord x divides n iff x in n -roots_of_1 )

proof end;

theorem Th35: :: UNIROOTS:35

for n being non zero Element of NAT holds n -roots_of_1 = { x where x is Element of (MultGroup F_Complex) : ord x divides n }

proof end;

theorem Th36: :: UNIROOTS:36

for n being non zero Element of NAT
for x being set holds
( x in n -roots_of_1 iff ex y being Element of (MultGroup F_Complex) st
( x = y & ord y divides n ) )

proof end;

definition

let n be non zero Element of NAT ;

func n -th_roots_of_1 -> strict Group means :Def3: :: UNIROOTS:def 3
( the carrier of it = n -roots_of_1 & the multF of it = the multF of F_Complex || (n -roots_of_1) );

existence
ex b₁ being strict Group st
( the carrier of b₁ = n -roots_of_1 & the multF of b₁ = the multF of F_Complex || (n -roots_of_1) )

proof end;

uniqueness
for b₁, b₂ being strict Group st the carrier of b₁ = n -roots_of_1 & the multF of b₁ = the multF of F_Complex || (n -roots_of_1) & the carrier of b₂ = n -roots_of_1 & the multF of b₂ = the multF of F_Complex || (n -roots_of_1) holds
b₁ = b₂ ;

end;

:: deftheorem Def3 defines -th_roots_of_1 UNIROOTS:def 3 :
for n being non zero Element of NAT
for b₂ being strict Group holds
( b₂ = n -th_roots_of_1 iff ( the carrier of b₂ = n -roots_of_1 & the multF of b₂ = the multF of F_Complex || (n -roots_of_1) ) );

theorem :: UNIROOTS:37

for n being non zero Element of NAT holds n -th_roots_of_1 is Subgroup of MultGroup F_Complex

proof end;

definition

let n be non zero Nat;
let L be non empty left_unital doubleLoopStr ;

func unital_poly (L,n) -> Polynomial of L equals :: UNIROOTS:def 4
((0_. L) +* (0,(- (1_ L)))) +* (n,(1_ L));

coherence
((0_. L) +* (0,(- (1_ L)))) +* (n,(1_ L)) is Polynomial of L

proof end;

end;

:: deftheorem defines unital_poly UNIROOTS:def 4 :
for n being non zero Nat
for L being non empty left_unital doubleLoopStr holds unital_poly (L,n) = ((0_. L) +* (0,(- (1_ L)))) +* (n,(1_ L));

Lm4: unital_poly (F_Complex,1) = <%(- (1_ F_Complex)),(1_ F_Complex)%>
;

theorem Th38: :: UNIROOTS:38

for L being non empty left_unital doubleLoopStr
for n being non zero Element of NAT holds
( (unital_poly (L,n)) . 0 = - (1_ L) & (unital_poly (L,n)) . n = 1_ L )

proof end;

theorem Th39: :: UNIROOTS:39

for L being non empty left_unital doubleLoopStr
for n being non zero Nat
for i being Nat st i <> 0 & i <> n holds
(unital_poly (L,n)) . i = 0. L

proof end;

theorem Th40: :: UNIROOTS:40

for L being non empty non degenerated well-unital doubleLoopStr
for n being non zero Element of NAT holds len (unital_poly (L,n)) = n + 1

proof end;

registration

let L be non empty non degenerated well-unital doubleLoopStr ;
let n be non zero Element of NAT ;

cluster unital_poly (L,n) -> non-zero ;

correctness
coherence
unital_poly (L,n) is non-zero ;

proof end;

end;

theorem Th41: :: UNIROOTS:41

for n being non zero Element of NAT
for x being Element of F_Complex holds eval ((unital_poly (F_Complex,n)),x) = ((power F_Complex) . (x,n)) - 1

proof end;

theorem Th42: :: UNIROOTS:42

for n being non zero Element of NAT holds Roots (unital_poly (F_Complex,n)) = n -roots_of_1

proof end;

theorem Th43: :: UNIROOTS:43

for n being Element of NAT
for z being Element of F_Complex st z is Real holds
ex x being Real st
( x = z & (power F_Complex) . (z,n) = x |^ n )

proof end;

theorem Th44: :: UNIROOTS:44

for n being non zero Element of NAT
for x being Real ex y being Element of F_Complex st
( y = x & eval ((unital_poly (F_Complex,n)),y) = (x |^ n) - 1 )

proof end;

theorem Th45: :: UNIROOTS:45

for n being non zero Element of NAT holds BRoots (unital_poly (F_Complex,n)) = ((n -roots_of_1),1) -bag

proof end;

theorem Th46: :: UNIROOTS:46

for n being non zero Element of NAT holds unital_poly (F_Complex,n) = poly_with_roots (((n -roots_of_1),1) -bag)

proof end;

theorem Th47: :: UNIROOTS:47

for n being non zero Element of NAT
for i being Element of F_Complex st i is Integer holds
eval ((unital_poly (F_Complex,n)),i) is Integer

proof end;

definition

let d be non zero Nat;

func cyclotomic_poly d -> Polynomial of F_Complex means :Def5: :: UNIROOTS:def 5
ex s being non empty finite Subset of F_Complex st
( s = { y where y is Element of (MultGroup F_Complex) : ord y = d } & it = poly_with_roots ((s,1) -bag) );

existence
ex b₁ being Polynomial of F_Complex ex s being non empty finite Subset of F_Complex st
( s = { y where y is Element of (MultGroup F_Complex) : ord y = d } & b₁ = poly_with_roots ((s,1) -bag) )

proof end;

uniqueness
for b₁, b₂ being Polynomial of F_Complex st ex s being non empty finite Subset of F_Complex st
( s = { y where y is Element of (MultGroup F_Complex) : ord y = d } & b₁ = poly_with_roots ((s,1) -bag) ) & ex s being non empty finite Subset of F_Complex st
( s = { y where y is Element of (MultGroup F_Complex) : ord y = d } & b₂ = poly_with_roots ((s,1) -bag) ) holds
b₁ = b₂ ;

end;

:: deftheorem Def5 defines cyclotomic_poly UNIROOTS:def 5 :
for d being non zero Nat
for b₂ being Polynomial of F_Complex holds
( b₂ = cyclotomic_poly d iff ex s being non empty finite Subset of F_Complex st
( s = { y where y is Element of (MultGroup F_Complex) : ord y = d } & b₂ = poly_with_roots ((s,1) -bag) ) );

theorem Th48: :: UNIROOTS:48

cyclotomic_poly 1 = <%(- (1_ F_Complex)),(1_ F_Complex)%>

proof end;

theorem Th49: :: UNIROOTS:49

for n being non zero Element of NAT
for f being FinSequence of the carrier of (Polynom-Ring F_Complex) st len f = n & ( for i being non zero Element of NAT st i in dom f holds
( ( not i divides n implies f . i = <%(1_ F_Complex)%> ) & ( i divides n implies f . i = cyclotomic_poly i ) ) ) holds
unital_poly (F_Complex,n) = Product f

proof end;

theorem Th50: :: UNIROOTS:50

for n being non zero Element of NAT ex f being FinSequence of the carrier of (Polynom-Ring F_Complex) ex p being Polynomial of F_Complex st
( p = Product f & dom f = Seg n & ( for i being non zero Element of NAT st i in Seg n holds
( ( ( not i divides n or i = n ) implies f . i = <%(1_ F_Complex)%> ) & ( i divides n & i <> n implies f . i = cyclotomic_poly i ) ) ) & unital_poly (F_Complex,n) = (cyclotomic_poly n) *' p )

proof end;

theorem Th51: :: UNIROOTS:51

for d being non zero Element of NAT
for i being Element of NAT holds
( ( (cyclotomic_poly d) . 0 = 1 or (cyclotomic_poly d) . 0 = - 1 ) & (cyclotomic_poly d) . i is integer )

proof end;

theorem Th52: :: UNIROOTS:52

for d being non zero Element of NAT
for z being Element of F_Complex st z is Integer holds
eval ((cyclotomic_poly d),z) is Integer

proof end;

theorem Th53: :: UNIROOTS:53

for n, ni being non zero Element of NAT
for f being FinSequence of the carrier of (Polynom-Ring F_Complex)
for s being finite Subset of F_Complex st s = { y where y is Element of (MultGroup F_Complex) : ( ord y divides n & not ord y divides ni & ord y <> n ) } & dom f = Seg n & ( for i being non zero Element of NAT st i in dom f holds
( ( ( not i divides n or i divides ni or i = n ) implies f . i = <%(1_ F_Complex)%> ) & ( i divides n & not i divides ni & i <> n implies f . i = cyclotomic_poly i ) ) ) holds
Product f = poly_with_roots ((s,1) -bag)

proof end;

theorem Th54: :: UNIROOTS:54

for n, ni being non zero Element of NAT st ni < n & ni divides n holds
ex f being FinSequence of the carrier of (Polynom-Ring F_Complex) ex p being Polynomial of F_Complex st
( p = Product f & dom f = Seg n & ( for i being non zero Element of NAT st i in Seg n holds
( ( ( not i divides n or i divides ni or i = n ) implies f . i = <%(1_ F_Complex)%> ) & ( i divides n & not i divides ni & i <> n implies f . i = cyclotomic_poly i ) ) ) & unital_poly (F_Complex,n) = ((unital_poly (F_Complex,ni)) *' (cyclotomic_poly n)) *' p )

proof end;

theorem Th55: :: UNIROOTS:55

for i being Integer
for c being Element of F_Complex
for f being FinSequence of the carrier of (Polynom-Ring F_Complex)
for p being Polynomial of F_Complex st p = Product f & c = i & ( for i being non zero Element of NAT holds
( not i in dom f or f . i = <%(1_ F_Complex)%> or f . i = cyclotomic_poly i ) ) holds
eval (p,c) is integer

proof end;

theorem Th56: :: UNIROOTS:56

for n being non zero Element of NAT
for j, k, q being Integer
for qc being Element of F_Complex st qc = q & j = eval ((cyclotomic_poly n),qc) & k = eval ((unital_poly (F_Complex,n)),qc) holds
j divides k

proof end;

theorem Th57: :: UNIROOTS:57

for n, ni being non zero Element of NAT
for q being Integer st ni < n & ni divides n holds
for qc being Element of F_Complex st qc = q holds
for j, k, l being Integer st j = eval ((cyclotomic_poly n),qc) & k = eval ((unital_poly (F_Complex,n)),qc) & l = eval ((unital_poly (F_Complex,ni)),qc) holds
j divides k div l

proof end;

theorem :: UNIROOTS:58

for n, q being non zero Element of NAT
for qc being Element of F_Complex st qc = q holds
for j being Integer st j = eval ((cyclotomic_poly n),qc) holds
j divides (q |^ n) - 1

proof end;

theorem :: UNIROOTS:59

for n, ni, q being non zero Element of NAT st ni < n & ni divides n holds
for qc being Element of F_Complex st qc = q holds
for j being Integer st j = eval ((cyclotomic_poly n),qc) holds
j divides ((q |^ n) - 1) div ((q |^ ni) - 1)

proof end;

theorem :: UNIROOTS:60

for n being non zero Element of NAT st 1 < n holds
for q being Element of NAT st 1 < q holds
for qc being Element of F_Complex st qc = q holds
for i being Integer st i = eval ((cyclotomic_poly n),qc) holds
|.i.| > q - 1

proof end;