Factorization of Cyclotomic Polynomials with Quadratic Radicals in the Coefficients

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1 Advances in Pure Mathematics, 07, 7, ISSN Onine: ISSN Print: Factoriation of Cycotomic Poynomias with Quadratic Radicas in the Coefficients Afred Wünsche Formery: Institut für Physi, Humbodt-Universität, Berin, Germany How to cite this paper: Wünsche, A. (07) Factoriation of Cycotomic Poynomias with Quadratic Radicas in the Coefficients. Advances in Pure Mathematics, 7, Received: August, 07 Accepted: September 4, 07 Pubished: September 7, 07 Copyright 07 by author and Scientific Research Pubishing Inc. This wor is icensed under the Creative Commons Attribution Internationa License (CC BY 4.0). Open Access Abstract In this artice we continue the consideration of geometrica constructions of reguar n-gons for odd n by rhombic bicompasses and ruer used in [] for the construction of the reguar heptagon ( n = 7 ). We discuss the possibe factoriation of the cycotomic poynomia in poynomia factors which contain not higher than quadratic radicas in the coefficients whereas usuay the factoriation of the cycotomic poynomias is considered in products of irreducibe factors with integer coefficients. In considering the reguar heptagon we find a modified variant of its construction by rhombic bicompasses and ruer. In detai, supported by figures, we investigate the case of the reguar tridecagon ( n = 3 ) which in addition to n = 7 is the ony candidate with ow n (the next to this is n = 769 ) for which such a construction by rhombic bicompasses and ruer seems to be possibe. Besides the coordinate origin we find here two points to fix for the possibe appication of two bicompasses (or even four with the addition of the compex conjugate points to be fixed). With ony one bicompass one has in addition the probem of the trisection of an ange which can be soved by a neusis construction that, however, is not in the spirit of constructions by compass and ruer and is difficut to reaie during the action of bicompasses. As discussed it seems that to finish the construction by bicompasses the correated action of two rhombic bicompasses must be appied in this case which avoids the disadvantages of the neusis construction. Singe rhombic bicompasses aow to draw at once two circes around two fixed points in such correated way that the position of one of the rotating points on one circe determines the positions of a the other points on the second circe in unique way. The nown case n = 7 embedded in our method is discussed in detai. Keywords Geometrica Constructions by Compass and Ruer, Bicompasses, Cycotomic Poynomias, Chebyshev Poynomias, 7-Gon, 3-Gon, 7-Gon, Fermat Numbers DOI: 0.436/apm Sep. 7, Advances in Pure Mathematics

2 . Introduction From ancient time on it was a probem of serious and of recreationa mathematics which of the reguar n-gons may be constructed by compass and ruer (straightedge without mars) and a simpe constructions were nown without a proof of the competeness of their possibiities up to the appearance of Gauss (in German: Gauß) on the scene at the very beginning of the 9-th century. Gauss showed that the basic numbers n for such constructions are the prime Fermat numbers F = +, ( = 0,,, ) with the ong nown cases corresponding to n= F0 = 3 and n= F = 5 and with the first unnown and surprising case at this time n= F = + = 7 (e.g., []-[3] and the more popuar artices of Gardner [4] [5]). This resuts from the soution of the cycotomic equations for these cases. A itte ater the genera theory was deveoped for the sovabiity of poynomia equations with integer or rationa coefficients in radicas (now caed Gaois theory) to which the cycotomic equation is a specia case. Some prehistory to this connected with names such as Lagrange, Ruffini and Abe is tod by Stewart [5] (chap. 8). The construction by compass and ruer requires not higher than quadratic radicas. In [] it was shown that the reguar heptagon ( n = 7 ) can be constructed by rhombic bicompasses and ruer. The rhombic bicompasses are two correated compasses with, at east, 3 connected arms of equa ength which can be fixed in two different points and which aow then the motion of the arms in two correated circes around the fixed points with one degree of freedom. The addition of such bicompasses as device for geometric constructions is, in our persuasion, certainy in the spirit of the ancient geometers and extends our possibiities for constructions. Exact constructions with rhombic bicompasses are possibe if the fixed points are determined by not higher than quadratic radicas (nested square roots) and, therefore, are constructibe by compass and ruer. To appy rhombic bicompasses and ruer for the construction of reguar n- n gons it is necessary that the cycotomic poynomias can be factoried into products of poynomia equations of 3-rd degree with not higher than quadratic radicas in their coefficients. In Sections 3, 9 and we suggest arguments that this restricts the possibe appications for odd n to prime numbers n equa to n= G = 3 + in anaogy to the Fermat numbers F that means to n = 7 for = 0 and next to n = 3 for = and discuss these cases in detai. In present artice we investigate the factoriations of the cycotomic poynomias for ow odd n (up to n = 9 ) in poynomia factors of 3-cyces (poynomia equations of 3-rd degree with 3 invoved roots) which contain not higher than quadratic radicas in the coefficients and expain how this can be obtained in expicit form. We aso give in expicit form for ow n the factoriation with ony quadratic radicas in the coefficients but with other than 3- cyces and determine some genera rues for this. For odd order DOI: 0.436/apm Advances in Pure Mathematics

3 n= m+, m=,, in a first step the factoriation of the cycotomic poynomia in a product of two poynomias of m-th degree with not higher than quadratic radicas in the coefficients is generay possibe. Concerning the reguar heptagon ( n = 7 ) we add a further modification of its construction by rhombic bicompasses and ruer and we discuss in detai the interesting case n = 3 where some probems remain open. For the we-nown case n = 7 we find in fuy expicit form the factoriation of the cycotomic poynomia in 8-4- and -cyces with nested quadratic radicas in the coefficients. For odd and even n we express the the poynomias for the determination of the Cosines of the anges of the circe-division probem by Chebyshev poynomias of first and second ind and derive more information about this in an appendix. The cycotomic poynomias pn ( ) for the compex corner points of a reguar n-gon to circumradius r = at the coordinate origin as it is we nown are n * p =, x+ i y x, y, x i y x, y. (.) n π n The n compex soutions = = exp i,( = 0,,,, moduo n) cycotomic equation n n n 0 n of the 0= = , =, (.) sove the probem of the circe division into n equa sectors and determine the corner points π * expi, =, n (.3) + = =, = 0,,,, mod n, + ( ) of the reguar n-gon in the compex pane. In the foowing we describe the procedure to obtain factoriations of the cycotomic equation with not higher than quadratic radicas in the coefficients and give the expicit resuts for odd n up to n = 9. In particuar, we discuss in detai the cases n = 7 and n = 3 which possess a reation to the appication of bicompasses and ruer. In the case n = 7 which we aso discuss in some detai we demonstrate how our method acts in a case nown since Gauss. The resuts for a corner points of the reguar 7-gon are given in an expicit form (see and compare aso [9] [] []).. The Cosine of the Anges for the Cycotomic Poynomias in Odd Case n= m+ We consider in this Section the case of the Cosine of the anges for the odd case n= m+, ( m=,, ) of the reguar n-gon and introduce the Cosine of anges for on the unit circe by * + + x = = cos ( θ ). (.) DOI: 0.436/apm Advances in Pure Mathematics

4 The cycotomic poynomias p ( ) p m+ ( ) ( ) m 0 + can be transformed to m m m m cos( θ) T ( cos( θ) ) = + + = + = + = = = m ( x) ( x) ( x) = + T = U + U, m m = where T n ( x ) are the Chebyshev poynomias of first ind and (.) U n x the Chebyshev poynomias of second ind (e.g., [6] [7] [8]). The we-nown property of the Chebyshev poynomias ( ( θ) ) ( nθ) n( ( θ) ) (( n + ) θ ) sin ( θ ) sin T cos = cos, U cos =, (.3) n is used. The reation between U n ( x ) and T n x in the second ine of (.) together with many other reations for the Chebyshev poynomias may be proved by compete induction using the addition theorems for the trigonometric functions. For a few first poynomias Um( x) + Um ( x) concerning the odd reguar n-gons ( n= m+ ) one finds expicity together with possibe factoriations with integer or rationa coefficients (i.e. in or ) in case of prime or composite n: One pecuiarity is that the poynomias Um( x) + Um ( x) tae on their simpest form with owest integer coefficients by the substitution x = u (simiary to the poynomias U m ( x ) themseves; but not T n ( x ) ). These poynomias possess factoriations with integer coefficients for composite numbers n m U + U are sometimes denoted by = +. The poynomias n n V n ( ) (see Appendix A). π The Cosines cos( θ ) = cos n of the anges θ are obtained from the soutions x = cos( θ ) of the equation u Tabe. Cycotomic poynomias for odd n and variabe x = cos( θ ) and integer factoriation. n m u u U U ; m + m u x = cos 0 3 u ( u )( u 3u ) ( θ ) u + u u + u u u + u u u+ u + u u u + u+ u + u u u + u + u u + u u u + u + u u u 8 7 = ( u )( u u )( u u 4u 4u ) = u 7u 6u + 5u + 0u 0u 4u u u 8u 7u u 5u 0u 0u 5u (.4) DOI: 0.436/apm Advances in Pure Mathematics

5 m+ m ( ) * p + = Um( x) + Um ( x) = 0, x cos ( θ ), (.5) with excusion of the soution 0 = x0 =. From the geometrica meaning of the poynomias as providing the doubed Cosines of the ange division probem as roots for odd n= m+ within the unit circe it is fuy obvious that a roots of a poynomias in (.4) possess ony m rea soutions within the imits x = u. Without further going into detais we mention that in case of even numbers n= m+ the cycotomic poynomias for the Cosines of the anges θ can be represented in the form (we use T ( x) = T ( x) ) m m m m p m m+ ( ) m + = cos( ( ) ) m = = m θ = 0 = 0 = 0 (.6) m m * + = Tm ( cos( θ) ) = Tm ( x) = U m( x), x cos ( θ). = 0 = 0 The vanishing of these poynomias provides as soutions the possibe Cosines of the anges to the corners of the n-gons with even n= m+ with excusion of the aready eiminated Cosines x0 =, x m + =. We mention here that as (irreducibe) cycotomic poynomias Φ n ( ) are n mosty understood the poynomias pn ( ) = divided by a products of (irreducibe) poynomias Φ d ( ) where d < n runs over a divisors of n (i.e., irreducibe in the sense of coefficients in or in but iey here aready in ) (e.g., van der Waerden [3], Stiwe [7], Shoni [] (p. 40)). They are for n paindromic poynomias (ony Φ ( ) = is not paindromic) with rea coefficients. Genera expicit formuae for the poynomias Φ n ( ) seem to be possibe for different divisibiity casses and for prime n= p it is pn ( ) p Φ n ( ) = = that means they are then of degree p. Stiwe [7] = 0 (p. 70) mentions as a curious property of the poynomias Φ n ( ) that the first poynomia Φ n ( ) with coefficients of moduus besides (and 0) is Φ 05 ( ) of degree 48 whereas a poynomias Φ n ( ) with n < 05 possess coefficients ony of moduus or equa to 0. In connection with constructions by rhombic bicompasses and ruer we are pn ( ) mainy interested in factoriations of the poynomias with not higher than (in genera, nested) quadratic radicas in the coefficients. A genera expicit m By definition, a poynomia P( m ) = a is paindromic if for a coefficients hods a = a = 0 m. With a PC and program Mathematica one may astonishingy easiy and quicy cacuate the poynomias Φ n n ( ) for high n in expicit form by factoriation of p ( ) = n. It seems that such poynomias with coefficients of moduus preferaby appear if the composite numbers n are products of primes 3, 5, 7,,... (e.g, n = 05 = ) but not for a such products (not besides for n < 05, e.g., for n = 3 but for, e.g., n = 65,95,385 ). For n = 55 = the poynomia Φ 55 ( ) contains a ot of coefficients of moduus,, 3 or equa to 0 and is of degree 480. DOI: 0.436/apm Advances in Pure Mathematics

6 formua for the factoried poynomias coud not be obtained but a procedure wi be described how such factoriation in given cases eads to the resut. 3. Cyces in the Circe-Division Probem We expain in this Section the factoriation of the cycotomic equation for odd n= m+ and concentrate us to the case n= p where p is a prime number π arger than p =. A soution = expi of the cycotomic equation n n = 0 is caed a primitive root if there does not exist a positive integer m π m< n for which =. It is cear that in each case = expi is a n primitive root and that for prime numbers n= p a soutions with =,,, n, ( mod n) are primitive roots (ony 0 = is never primitive). pn ( ) For prime n= p the cycotomic poynomia in the form cannot be factoried into poynomias with integer coefficients. It is said that it is irreducibe in and therefore aso in with coefficient of the highest power equa to. Obviousy, this does not mean that it is not factoriabe into poynomias with radicas in the coefficient from which such are interesting for us which contain not higher than quadratic radicas since such radicas are constructibe by compass and ruer. To obtain such factoriations one may appy a procedure using the itte theorem of Fermat. We expain this in the foowing. According to the itte theorem of Fermat (e.g., [3] [8] [9]) for prime numbers p and natura numbers g =,,, ( mod p) hods (symbo stands here for congruences moduo p) ( p) p g, mod. (3.) This means that for prime numbers n= p and for the primitive root we have n n n g g g = =, = = =. (3.) n g We choose positive integers g =,,, n and form first with the soution 0 g g g g = =, =, =,, = up to the case the sequences ( 0 g g g g ) + g, mod + g n when = =. Each such sequence we ca a cyce. The sequence (,,,, ) n g g g, atest after the next step to n, eads bac to the initia soution. If is a divisor of n = m then, depending on the choice of g, the mentioned sequence may ead bac to aready after + steps. For odd prime n= m+ the numbers =, = m and = m are divisors of n = m and there are sequences with cyce engths beonging to the choice g = m and such with cyce engths m and m. If one does not begin with the eement in a cyce but with another eement of this cyce then one obtains by the described procedure the same cyce with rotated order of the eements. If a root is not contained in a certain cyce then we form the g DOI: 0.436/apm Advances in Pure Mathematics

7 0 g g g g sequences ( = 0 =, = g, =,, = g g g ) and obtain for prime n an equivaent cyce of the same ength and so we may continue up to the case when a numbers 0, ( mod n are comprised. We iustrate the factoriation in cyces for our two most interesting prime cases n = 7 and 3:. Case n = 7, basic cyces (a equaities are moduo 7 in the indices) g = :, =, g = :,,, =, 4 8 g = 3:,, =,, =, =, =, g = 4:,, =, =, g = 5:,, =, =, =, =, =, g = 6:,, =. (3.3) 6 36 For g = 3 and g = 5 the sequences are 6-cyces which comprise a n soutions 0 = in different order of the eements. For g = and g = 4 we find equivaent 3-cyces in different order of the eements (,, 4) can be compemented by the 3-cyce ( 3, 6, 5 ) 0. For g = n = 6 we find the -cyce (, 6) compemented by the other possibe -cyces (, ) and (, ) which = to comprise a eements 5 which can be for which one does not find a factoriation in poynomias of -nd degree with ony quadratic radicas in the coefficients. In the trivia case g = one finds in every case ony the -cyce with eement ( ) which can be compemented by -cyces ( ) of the other roots.. Case n = 3 3 4, basic cyces (a equaities are moduo 3 in the indices) g = :, =, g = :,,,, =,,, =, =, =,, =, =, g = 3:,,, =, g = 4:,, =,, =, =, =, g = 5:,, =, =, =, g = 6:,, =, =, =, =,, =, =, =, =, =, =, g = 7:,, =, =, =, =, =, =, =, =, =, =, =, g = 8:,, =, =, =, g = 9:,, =, =, g = 0:,, =, =, =, =, =, DOI: 0.436/apm Advances in Pure Mathematics

8 g = :,, = 4, 44 = 5, 55 = 3, 33 = 7, 77 =, 3 =, =, =, =, =, =, g = :,, =. (3.4) 44 We have here cyces of engths,,3, 4,6, which a are divisors of n = m=. For exampe, we find the foowing four 3-cyces {(, 3, 9 ),( 4, 0, )},{(, 5, 6 ),( 7, 8, )} covering a primitive roots where the two pairs of 3-cyces in braces form two 6-cyces. The 3-cyces foow from the subdivision of the -cyces in two step by division of eading first to two 6-cyces and then in ast step by division of to four 3-cyces. The subdivision of the cycotomic equation of -th degree in a product of six - cyces with quadratic equations containing 6 paired roots * (, = 3, ( =,,, 6) ) does not ead to the expicit form of the quadratic equations since the resoution of the -degree cycotomic poynomia in one step by division of 3 is not possibe with coefficients in form of quadratic radicas independenty from the order in which the division by 3 is made, from 4 or 6. Therefore the three 4-cyces (, 5, 8, ),(, 3, 0, ),( 4, 6, 7, 9 ) obtained from choice g = 5 and g = 8 are aso not to find in form of poynomia equations with ony quadratic radicas as coefficient. Each of these three 4-cyces contains ony one of the roots of the four 3-cyces. If we oo to the cyces in (3.3) and (3.4) we find in case of g for the sum of the powers of g within the cyce + j g g = n, mod n, ( g ). (3.5) g j= 0 This is a genera property which we wi prove now. According to the definition of a cyce of ength the power g is the ast in the cyce before the next power g + eads bac the root to + = and g + is congruent to g moduo n. Since the sum on the eft-hand side in (3.5) is a positive integer and g 0 is aso a positive integer the right-hand side in (3.5) is a positive integer and due to the given congruence a mutipe of n. From this foows for the product of roots within a cyce with the primitive root j + g g j= 0 g j g + g g g g 0 j= 0 g = = = = = =. (3.6) The same is the case with each cyce of the ength containing an arbitrary primitive root. A consequence is that we now at once the constant term in the factoried cycotomic poynomias that is of importance when we begin from behind (ow-order -terms proportiona to ) to find the factor poynomias in factoriations. The question about the cycotomic poynomias pm+ ( ) of degree m DOI: 0.436/apm Advances in Pure Mathematics

9 Tabe. Fermat numbers numbers. F and reated numbers G and factoriation into prime F + prime factors G + prime factors (3.7) which can be spit into products of poynomias of 3-rd degree (3-cyces) eads basicay, anaogousy to Fermat numbers F = +, to numbers G 3 = + which have to be prime numbers. Tabe shows the few initia possibiities up to = 5. For the next three cases = 6,7,8 the numbers F and G are composite numbers as the computer shows but these numbers grow very fast and my PC (with program Mathematica 0 ) did not provide a resut for the next case = 9 of both numbers in acceptabe time. However, it is now nown that a numbers F from = 9 up to = 9 are composite without nowing a prime factors in a these cases (see [9], end of chap. 5). Since the number G = 49 is composite it is not a possibe candidate for construction of the reguar 49-gon by rhombic bicompasses and ruer. We mention here that if one admits ange trisection by a neusis construction attributed to Archimedes [] [9] as an additiona eement of constructions which in our persuasion is not in the spirit of ancient constructions by compass and ruer then one comes to possibe numbers for the soubiity of the circe division probem of the form P, = 3 +, (, = 0,,, ) if they are prime numbers and which are caed Pierpont numbers (from 895, see [9] [0]). These numbers are more genera ones than the Fermat numbers F and aso than the numbers G in Tabe (see aso end of Section ). 4. Factoriation of Cycotomic Poynomia for n = 3 with Rea Coefficients In case of n = 3 the cycotomic poynomia ( ) p3 can be represented 3 i 3 + i 3 = + + = + + = ( )( ). It is written down here for the anaogy to higher ess trivia cases. 5. Factoriation of Cycotomic Poynomia for n = 5 with Quadratic Radicas In case of n = 5 the cycotomic poynomia with two -cyces which has the form ( ) p5 (4.) possesses the factoriation DOI: 0.436/apm Advances in Pure Mathematics

10 5 = = = (( )( 4) )(( )( 3) ) The compex conjugate roots (, ) and (, ) (5.) 4 3 moduo 5 are here paired in one of the two -cyces and therefore the coefficients in form of quadratic radicas possess rea vaues. The case of the reguar pentagon is aso commony nown and we do not consider it in detai. 6. Factoriation of Cycotomic Poynomia for n = 7 with Quadratic Radicas p7 In case of n = 7 the cycotomic poynomia possesses the foowing factoriation with two 3-cyces with ony quadratic radicas in the coefficients 7 = ( ) (( )( )( 4) )(( 3)( 5)( 6) ) 3 i 7 + i i 7 i 7 = + + =. (6.) The factoriation in this case with ony quadratic radicas in the coefficients and concerning the corners of the reguar heptagon was discussed in []. It is easy to determine this factoriation with quadratic radicas in the coefficients from 4 i * * * + + = + y and = = i y with imagi- 6 4 nary party to find from the poynomia using = = and 4= + + =. The primitive roots, ( =,, 6) of the poynomias (6.) form two 3-cyces (,, 4) and ( 6, = 5, 4 = 0 = 3 ) moduo 7 in the indices. The quadratic radica for the sum together with the circe division in 7 equa parts is shown in Figure. In Figure we iustrate possibiities for the construction of the reguar heptagon by bicompasses and ruer and draw additionay a circe with radius r = around the coordinate origin ( 0,0 ) which is equa to the distance 7 from the coordinate origin to the fixed point, of the bicompasses. This circe possesses intersection points with the circe of radius r = around + i 7 the mentioned second fixpoint at + = i 7 = and at 5+ i 7 = in the compex pane as one easiy cacuates. In this way we see 4 that the intersection point + i 7 ies on the second arm of the absoute mini- DOI: 0.436/apm Advances in Pure Mathematics

11 Figure. Reguar heptagon with axes projection of corners in compex -pane. Besides the coordinate origin = 0 the second fixed point for the rhombic bicompasses is + i 7 = (figure from [] made by Mathematica 6 ). mum 3-arm bicompasses (see []) in the right position for the construction of the reguar heptagon that means on the ine between and +. Ceary, one has not ony to beieve to the optica impression but have to prove it. The ine between and + can be parameteried by * = + r, 0 r, r = r, (6.) with rea parameter r. For the parameter vaue r = (numericay r ) one finds that the vaue on the ine between and + coincides with the point + i according to 4 * * r = = + + = + + = r, i 7 = + = =. (6.3) 4 4 Thus we have to bring the second arm of the absoute minimum 3-arm bicompasses in the position that it intersects the point 7, which ast can DOI: 0.436/apm Advances in Pure Mathematics

12 be constructed by the intersection of the two mentioned circes and an aternative method of construction of the reguar heptagon is described. + i 7 In addition, Figure shows that the point = ies aso on the proongation of the ine between + 4 and. This ine can be parameteried with rea parameter s by * = + s, 0 s, s = s. (6.4) For the parameter vaue s = (numericay s ) one 4 finds that the vaue on the proongation of the ine between + 4 and Figure. Reguar heptagon and construction with rhombic bicompasses and ruer in compex -pane. Additionay to the fixed points for the bicompasses and corresponding circes of radius r = we have drawn a circe around the coordinate origin with radius r = and obtain in this way a modified construction by absoute minimum bicompasses with 3 arms (figure made as a foowing figures by Mathematica 0 ). DOI: 0.436/apm Advances in Pure Mathematics

13 coincides with the point + i according to 4 * * s = = = = s, i 7 = + = =, (6.5) that affirms the mentioned intersection. In [] it was aready shown that the point = ies on the ine between 4 and + 4 which can be parameteried by * = + t, 0 t, t = t, (6.6) with rea parameter t. With the parameter vaue t = (numericay t = ) foows + ( ) 4 * * t = = + = t + = = =, (6.7) that proves the statement. This, aternativey, can be aso used for the con- struction of the reguar heptagon by rhombic bicompasses and ruer. We mention that the parameters with the notation r and t are essentiay the same since r+ t =. * The equation for the Cosines u = + = x= cos( θ ) (equation for n = 7 in Tabe, Equation (.4)) u u 3 U3 + U = u + u u = 0, (6.8) is a 3-rd degree equation which cannot be soved ony in quadratic radicas as it is nown and its soution invoves (compex) cubic radicas. Therefore, this does not hep for the construction by compass and ruer. However, with other means of construction (e.g., neusis construction of ange trisection [] [9] [4] and Geason [3] (ange p-section)) this becomes possibe. Thus the reguar heptagon oses a itte its horror as not constructibe by compass and ruer between the cases of the reguar trigon n = 3 and the reguar octagon n = 8 since it is constructibe by bicompasses and ruer. 7. Factoriation of Cycotomic Poynomia for n = 9 in Different Ways Since n = 9 is a composite number we find different favorabe factoriations of the cycotomic poynomias. As specia roots the circuar division of the unit circe in 9 equa sectors contains the third roots of unity and we have the factoriation, DOI: 0.436/apm Advances in Pure Mathematics

14 9 = ( ){ } = i 3 + i 3 i 3 + i = { } ( 3)( 6) (( )( 4)( 7) )(( )( 5)( 8) ) =, (7.) where the second factor in braces is of the form of the first factor with the 3 substitution. It requires the ange trisections of the anges to the roots 3 + i 3 3 i 3 * 3 = = and 6 = = = 3. This is possibe by the neusis construction nown from ancient time [] [9] which, however, is not in the spirit of constructions by compass and ruer. The ast is impossibe for amost a arbitrary anges incuding the ange π 3. Therefore, as nown, the circe division probem in case of n = 9 cannot be soved by compass and ruer since ± i 3 the third root of an arbitrary compex number (here of 3,6 = ) cannot be constructed in this way. This can be aso seen from the equation of Tabe π (Equation (.4)) for the doubed Cosines u = cos, 9 the soutions of the foowing poynomia equation in factoried form u u 3 U4 + U3 = ( u+ )( u 3u+ ) = 0. ( =,,, 8) as From the coefficients of the vanishing cubic poynomia foows π 4π 8π π 4π 8π cos + cos + cos = 0, cos cos cos =, (7.) (7.3) where the second reation does not provide independent in formation in comparison to the first. The cubic equation u 3 3u+ = 0 can be soved by cubic but not by quadratic radicas aone. The standard factoriation into two poynomias of 4-th degree is 9 4 i i i 3 3 i 3 = = {( 6) (( )( 4)( 7) )} ( 3) (( )( 5)( 8) ) { }. (7.4) In the factor poynomias of 4-th degree are contained two 3-cyces (, 4, 7) and (,, ) of unity paired with one of the third roots 3 and 6 Other genuine than 3- and 6-cyces do not exist in case of n = 9 but it happens that the root 0 = appears in the determination of the cyces according to the genera procedure (it is then no more a cyce) that for prime n is impossibe. A simiar interestingy simpe factoriation by two poynomias of 4-th degree foows directy from (7.) by the product DOI: 0.436/apm Advances in Pure Mathematics

15 9 4 i i i 3 3 i 3 = + ( + ) + ( + ) { } {( 3) (( )( 4)( 7) )} ( 6) (( )( 5)( 8) ) =. (7.5) and means the exchange of the factors 3 and 6. It is possibe due to n = 9 as a composite number. 8. Factoriation of Cycotomic Poynomia for n = with Quadratic Radicas In case of n = one has ony the foowing factoriation by 5-cyces which eads to poynomias with quadratic radicas in the coefficients = i i = i 4 3 i + + (( )( 3)( 4)( 5)( 9) ) ( ( )( 6 )( 7 )( 8 )( 0 )) = The 5-cyce in the first factor is formed by the roots (, 3, 9, 7 5, 5 4, ( )) conjugate roots (, 6, 8 7, 0, 30 8, ( 4 )) * The equation for the doubed Cosines u x cos( θ ). (8.) = = = and the second factor by the compex = = = =, a (mod ). = + = = (case n = in Tabe (Equation (.4)) as a genuine 5-th order equation without specia symmetries is not possibe to sove in radicas as it is nown. 9. Factoriations of Cycotomic Poynomia for n = 3 with Quadratic Radicas The case n = 3 is very interesting due to factoriation of the cycotomic equation by poynomias of 3-rd degree with ony quadratic radicas in the coefficients in 3-cyces. This maes it possibe for the appication of the rhombic bicompasses and ruer for the construction of the reguar tridecagon (3-gon). The first factoriation by two 6-cyces provides two factor poynomias of 6-th degree with rea coefficients = = = {( )( 3 )( 4 )( 9 )( 0 )( )} { ( )( 5)( 6)( 7)( 8)( ) } =. (9.) DOI: 0.436/apm Advances in Pure Mathematics

16 Each of the two factor poynomias of 6-th degree can be again factoried in two poynomias of 3-rd degree with ony quadratica radicas in the coefficients and the invoved eros we find from the four 3-cyces with the first invoving the eement expicity given in (3.4) 3 3 i i = i ( ) 3 i ( ) i ( ) 3 i ( ) 3 (9.) i ( ) + 3 i ( ) = {(( )( 3)( 9) )(( 4)( 0)( ) )} (( )( 5)( 6) )(( 7)( 8)( ) ) { } This means that the fixed points of the two rhombic bicompasses besides the coordinate origin = 0 are the foowing quadratic radicas = and = or their compex conjugates ( ) * * * 3 + i = = i0.546, = , , * * * 3 + i = = i.754, = , (9.3) Together with the circe division probem the two principay possibe fixed points (without the compex conjugate ones) for the bicompasses are shown in Figure 3. The sums and differences of the fixed points are aso expressibe by ony quadratic radicas. We find for the sums agebraicay and numericay + i = i.4784, ( ) ( ) DOI: 0.436/apm Advances in Pure Mathematics

17 Figure 3. Two variants of rhombic bicompasses in case of a reguar 3-gon. The two fixed points of the rhombic bicompasses are = 0 and = + + in first picture and 3 9 = 0 and = + + in second picture (see text) =.3078, ( ) ( ) i = 0.5 i.030, ( ) ( ) and for the differences =.3078, (9.4) ( ) ( ) i =.8078 i.030, ( ) ( ) =.673, ( ) ( ) i = i.4784, ( ) ( ) = (9.5) ( ) ( ) A these numbers and radiuses are constructibe by compass and ruer. The points,( 0 mod 3) aone and combinations of ony two such points in form of + are not constructibe by compass and ruer since they contain cubic radicas. This is iustrated in Figure 4 and Figure 5 where the two additiona possibiities with the compex conjugate fixed points are obtained by refection of each partia picture at the horionta ine through the coordinate origin. This provides 4 possibiities for the common combination of the two rhombic bicompasses where two of them are obtained again by refection on the horionta ine through the coordinate origin. The two essentiay different ones DOI: 0.436/apm Advances in Pure Mathematics

18 Figure 4. First of two variants of correated rhombic bicompasses in case of a reguar 3-gon. are iustrated in Figure 4 and Figure 5. According to Tabe (Equation (.4)) the doubed Cosines π π u = cos = cos( 3 ), 3 3 degree poynomia equation with 6 rea-vaued soutions u u U6 + U5 = u + u 5u 4u + 6u + 3u = 0. ( =,,, 6) are the roots of the 6-th (9.6) The poynomia on the eft-hand side of this equation can be factoried in two poynomias of 3-rd degree with ony quadratic radicas in the coefficients in the foowing way [3] u u u u u u = u u u+ u + u u This corresponds to the foowing reations for sums of the Cosines π π cos = cos ( 3 ) 3 3 (9.7) DOI: 0.436/apm Advances in Pure Mathematics

19 Figure 5. Second of two variants of correated rhombic bicompasses in case of a reguar 3-gon. π 6π 8π 3 cos + cos + cos =, π 0π π 3 + cos + cos + cos =, (9.8) and for products of the same Cosines π 6π 8π 3 3 cos cos cos =, π 0π π cos cos cos =. (9.9) One may obtain these reations from reations (9.3) for the fixed points of the rhombic bicompasses. Reations (9.9) do not give additiona independent information to reations (9.8). This can be seen if one transforms the products DOI: 0.436/apm Advances in Pure Mathematics

20 in (9.9) into sums using addition theorems for trigonometric functions. The same is true for the reations obtainabe from the coefficients in front of u in (9.7). Equation (9.6) together with (9.7) means that the cycotomic equation for n = 3 can be exacty soved by not higher than cubic radicas. This, however, is not appropriate for the construction by compass and ruer which aows ony quadratic radicas. The same concusion can be drawn for the fu soutions of the corner points of the reguar 3-gon by setting equa to ero the four 3-rd degree poynomias in (9.) of the factored cycotomic poynomia (9.). The rea construction of the reguar 3-gon by rhombic bicompasses and ruer seems to be rather compicated. One may distinguish two principa cases:. The use of ony one of the two possibe bicompasses shown in the two pictures in Figure 3.. The use of the two rhombic bicompasses in correated way in one of the two principay different variants as iustrated in Figure 4 and Figure 5. The probem in first case is that during the appication of the bicompasses, for exampe, to points and 3 in first partia picture in Figure 3 one has to find the position when the ange between and 3 is the doubed ange between 0 = and that is equivaent to sove the trisection of the ange between 0 and 3. The same probem arises if we use the points and 9 or 3 and 9 instead of and 3. In principa, an ange trisection can be made by a neusis construction attributed as aready said to Archimedes [] [9] but it is not in the spirit of construction by compass and ruer and, furthermore, it is uncear how it coud be combined at the same time with the action of the bicompasses. We coud not find a possibiity aso in case if we use in addition to one of the bicompasses the corresponding bicompasses with the compex conjugate fixed point. The second case with correated bicompasses seems to be, in principa, possibe. In case of the reguar heptagon we found points from the pictures on the arms of the bicompasses or on their proongation which can be expressed by not higher than quadratic radicas and proved this property then agebraicay. In case of the 3-gon we go a simiar way but start from an opposite point of view. We oo for possibiities of points on the ines between corners of bicompasses which can be represented by not higher than quadratic radicas. We consider the bicompasses which in the right position determine the corner points (, 3, 9) of the reguar 3-gon (see Figure 4). A first possibiity is then to oo on the ine between and + 6 with 6 on the other bicompasses which can be parameteried with rea parameter r by * = + r, r = r. (9.0) λ6 With the choice of a rea parameter r = with rea parameter λ 6 we find DOI: 0.436/apm Advances in Pure Mathematics

21 + + λ = = + + λ =, λ = λ, * * * r 3 3 r 6 ( ) + + λ 3 + i = + = λ = + λ, λ = (9.) The points λ 6 with rea λ 0 are, in genera, and contrary to not representabe by ony quadratic radicas and, therefore, we set λ = 0 in (9.). An anaogous second possibiity to the described one is to oo on the ine between 3 of the first bicompasses and with 5 of the second bicompasses which can be parameteried with rea parameter s by * = + s, s = s. (9.) Then with the choice of parameter s = we find + 9 * * s = = = s, = + = + +. (9.3) A third equivaent possibiity is to oo on the ine between 9 on the first bicompasses and 9 + with on the second bicompasses. It can be parameteried with rea parameter t by + 3 With the choice t = we obtain * = + t, t = t. (9.4) 9 + = = + = 3 * * t t, + = + = + +. (9.5) A three possibiities ead to the fixed point of the first bicompasses expressibe by quadratic radica through which or through their proongations the considered parameteried ines have to go. There are aso further equivaent possibiities to use instead of the constructibe points and their sums and differences which we did not investigate up to now in detai. For the rea construction one has to estabish a correation between the two bicompasses with fixed points ( 0, ) and ( 0, ) in such a way that when one of these bicompasses is in the right position the second at the same time has aso to be in the right position for the ange division in thirteen equa parts. This means that we have to guarantee that the second of the bicompasses acts in concerted way with the first. It seems that one can use for DOI: 0.436/apm Advances in Pure Mathematics

22 such a couping, for exampe, that the point of the first bicompasses ies exacty on the ine (arm) between = 0 and 6 of the second bicompasses (see Figure 4). This is aready cear from symmetry and does not need to be proved. One has to guarantee that the point equivaent to in the right position of the bicompasses may gide aong the ine from = 0 to 6 in the right position during the action of the two bicompasses. Then the point of the second bicompasses maes a bisection of the ange formed by the points and 3 of the first bicompasses during their action when they arrive the right position. To determine this bisection seems to be possibe during the action of the bicompasses. There are equivaent possibiities to reaie the construction by the two bicompasses. A wea point is the giding of a point (here 3 + 9) on a ine (here between = 0 and 6 ) as an admissibe action in the spirit of geometric constructions from ancient time on. This is somehow probematic and requires more discussion in future as we are abe to give here at this time. There are aso some not exact coincidences ooing onto the figures one coud thin to be exact coincidences and which in the study of anguages woud be caed fase friends. They may be used for approximate constructions of the 3-gon but it is not our intention to find and discuss them in detai. We mention ony a few ones which are evident from the figures. The point ies on the circe with radius equa to around the coordinate origin but not at the same time on the circe with radius equa to around that is aready opticay to sense in Figure 4 and Figure 5. The projection of the point + 3 onto the rea x-axis is not exacty equa to as it seems to be from Figure 4 and numericay we find i where the deviation of the rea part from is widey above the numerica errors of cacuation and it is nothing more to prove in this case. This is aso cear since + 3 is not representabe by quadratic radicas. The ine from coordinate origin 0 to fixed point does not bisect the ange between 4 and 5 as from para- 9π meteried ine set equa to the bisected ange t( ) = expi 3 with numericay cacuated compex (but not rea) t = i foows. 0. Factoriation of Cycotomic Poynomia for n = 5 with Quadratic Radicas According to the composite number n = 5 the cycotomic equation contains the cases of the cycotomic equations for n = 3 and n = 5 as factors and possesses the form 5 4 = = = ( 4 )( 3 8 )( ) = 0 (( 5 )( 0 ))(( 3 )( 6 )( 9 )( )) ( ( )( )( 4 )( 7 )( 8 )( )( 3 )( 4 )). (0.) DOI: 0.436/apm Advances in Pure Mathematics

23 The ast factor is the irreducibe part of the cycotomic poynomia pn ( ) (with coefficients in ) and is denoted by Φ can be decomposed in two compex conjugate factors according to Φ 5 = i 5 i = + i 5 + i (( )( )( 4)( 8) ) ( ( 7 )( )( 3 )( 4 )) = In this factoriations we have two 4-cyces with the eros (,, 4, 8) (,,, ) with (0.) and = =. From the given factoriations and from other considerations foow some nown possibiities to sove the circe division probem for n = 5 with compass and ruer and we do not further discuss this.. Factoriations of Cycotomic Poynomia for n = 7 with Quadratic Radicas The possibiity of the soution of the circe division probem in case n = 7 together with the soution of the genera probem from ancient time for which n it can be soved at a by compass and ruer was discovered by the young Gauss and is represented in numerous boos (e.g., [7] [9] [] and others). The representation here of this nown case iustrates our approach and faciitates its understanding in the unnown cases, in particuar for n = 3. A first factoriation of the cycotomic poynomia which is paindromic in two paindromic poynomia factors with quadratic radicas in the coefficients is 7 6 = = = ( ) ( ) ( ) ( 7 ) ( ) ( ) ( ) ( 7 ) {( )( )( 4 )( 8 )( 9 )( 3 )( 5 )( 6 )} { ( 3 )( 5 )( 6 )( 7 )( 0 )( )( )( 4 )} = The first factor, for exampe, invoves the eros. (.) DOI: 0.436/apm Advances in Pure Mathematics

24 ,,,,,, moduo7. The two paindromic factor poynomias of 8-th degree can be decomposed each in the product of two poynomias of 4-th degree which are aso paindromic in the foowing way: first factor ( + ) + ( + ) + ( + ) + ( + ) + ( 7 ) 7 ( 7 7 ) 7 7 ( ) 4 ( 3 ) = ( + ) (( )( 4 )( 3 )( 6 ))(( )( 8 )( 9 )( 5 )) =, (.) second factor ( + ) + ( + ) + ( + ) + ( + ) + ( + 7 ) 7 ( 7 7 ) 7 7 ( ) 4 ( 3 ) = ( + ) (( 3 )( 5 )( )( 4 ))(( 6 )( 7 )( 0 )( ) ) =. (.3) In the factoriation we have a product of 4 paindromic poynomias of 4-th degree each of which can be decomposed again in a product of paindromic poynomias of now -nd degree with ony quadratic radicas in the coefficients according to the genera formua = + + a a 4b a+ a 4b+ 8 a b a (.4) This provides the further spitting in products of pairs of two paindromic poynomias of -nd degree as foows: first factor 3 A paindromic poynomia of 8-th degree cannot be decomposed, in genera, in a product of paindromic poynomias of 4-th degree with not higher than quadratic radicas in the coefficients such as in formuae (.) and (.3) and this is ony possibe under certain restrictions to the coefficients of the 8-th degree paindromic poynomia. There are, however, aso some interesting reatives to paindromic poynomias of n-th degree without specia names. These are, in particuar, poynomias n where the coefficients in front of are equa to ( ± ) to that of or where they are compex conjugate to the coefficients in front of with possibe additiona factors ( ± ). DOI: 0.436/apm Advances in Pure Mathematics

25 ( )( 4 )( 3 )( 6 ) = (( )( 6 ))(( 4 )( 3 )) = second factor ( )( 8 )( 9 )( 5 ) = (( )( 5 ))(( 8 )( 9 )) = third factor ( 3 )( 5 )( )( 4 ) = (( 3 )( 4 ))(( 5 )( )) = fourth factor ( 6 )( 7 )( 0 )( ) = (( 6 )( ) )(( 7 )( 0 )) = ,,,. (.5) (.6) (.7) (.8) The bigger bracets in the short and in the expicit expressions correspond to each other in their ordering in the written form of the formuae. The coefficients in a the poynomias of second degree in (.5)-(.8) in front of are rea ones and the haf of the negativey taen vaue provides the DOI: 0.436/apm Advances in Pure Mathematics

26 Cosine of the corresponding ange. From first factor in first poynomia in (.5), π for exampe, one find cos, that means the Cosine of the ange to the first 7 corner of the 7-gon which is expicity π cos =. (.9) 7 6 This is identica in content (but not in form) with the expression derived by Gauss in Section 365 of Disquisitiones Arithmetic as given in the citation by Edwards [6] on p. 3 and here cited according to him (see aso Stewart [5] (p. π 3)). A vaues for cos, ( =,,,8) can be taen from the corres- 7 ponding coefficients in the quadratic factor poynomias in (.5)-(.8). Let us give for competeness and for the factoriation of the poynomia u u U8 + U7 for n = 7 in Tabe (Equation (.4)) with coefficients in form of quadratic radicas the compete expressions for the doubed Cosines for a primitive roots u. From (.5) and (.6) foows for π π u cos = cos( 7 ) 7 7 u u,4, ± = ± =, (.0) 8 where the first of the two indices in u corresponds to the upper sign and the, second to the ower one in ± and, anaogousy, from (.7) and (.8) u 3, ± = ± u6,7 =. (.) 8 6 The factoriation of taen together with (.5)-(.8) can be formay = 0 written = = ( u +, ) = 0 = π u cos, ( ( mod7 )), 7 where the expicit form of the coefficients u, ( =,,, m= 8),, (.) given in (.0) and (.) are taen from (.5) and (.8). If we insert the expressions u for u into the cycotomic Equation (.5) for m= 8, n= m+ = 7 using Tabe DOI: 0.436/apm Advances in Pure Mathematics

27 (Equation (.4)) u u U8 + U7 = u + u 7u 6u + 5u + 0u 0u 4u+ p ( ) ( ) ( u ) =, + = +, 7 * 8 (.3) then its vanishing is identicay satisfied. We checed this agebraicay by computer with Mathematica 4 and we checed aso numericay by computer that the order of the braces and big bracets in different parts of the Formuae (.) and (.3) and in Formuae (.5)-(.8) remains preserved. Since in case of the reguar 7-gon a Cosines of the anges are expressed by quadratic radicas the mutipication of arbitrary factors ( u u)( u u) eads again to poynomias with ony quadratic radicas in the coefficients. Therefore, the factoriation of cycotomic poynomias such as (.3) with quadratic radicas in the coefficients admits many possibiities but ony a few ead to simpe expressions. In a first step one may obtain the foowing factoriation of the poynomia of 8-th degree on the right-hand side of (.3) into the product of two poynomias of 4-th degree with quadratic radicas as coefficients = u + u u + ( + 7 ) u u + u u + ( 7 ) u u u u u u u u u (.4) This is certainy the simpest of the possibe factoriations of the 8-th degree poynomia in two poynomias with quadratic radicas as coefficients. The first factor poynomia contains the factors ( u u, u u4, u u, u u8) and the second the remaining factors ( u u3, u u5, u u6, u u7). Further factoriations of the poynomias of 4-th degree into products of poynomias of -nd degree with quadratic radicas in the coefficients are to obtain in anaogous way using Formuae (.)-(.3) and since each factoriation in factors u u contains ony quadratic radicas this is even possibe in different ways. We do not write down a this. The genera possibiities for the construction of reguar n-gons by compass and ruer according to the rues of ancient time which young Gauss finay found and proved and which the probem soved forever are the foowing (e.g., [7] [9]). The basis form such Fermat numbers F + which are prime numbers p i. We denote them by p, p,. An n-gon is constructibe by compass and ruer if it is a power (for ange bisections) mutipied by an arbitrary product of distinct prime Fermat numbers p i that means n= p p. (.5) i i 4 The used program Mathematica 0 does not automaticay provide the soutions of (.3) in the expicit form of the u but affirms immediatey the satisfaction of the equation if we insert there separatey their expicit vaues. DOI: 0.436/apm Advances in Pure Mathematics

28 As it is we nown the first 5 Fermat numbers which are F0 = 3, F = 5, F = 7, F3 = 57, F4 = and are a prime numbers but the next Fermat number F 5 is composite (e.g., [7] [9]; see aso Tabe in Section 3). The anaogous case of numbers G = 3 + for geometric constructions by bicompasses and ruer are given aso in Tabe. By combinations of constructions by compass and ruer with constructions by bicompasses and ruer one finds further possibiities to sove the circe division probem geometricay. For exampe, by combination of the constructions of the π π anges or with the anges π π or by bicompasses one may construct π π π π ( 7 3π ) π π ( 7 5π ) 35 = =, = =, 3 7 π π 0 35 π 3 π 0 π π ( 3 3) π 39 π π ( 3 5) π 65 = =, = =. (.6) π π We see that in connection with (repeated) ange bisections admitted by powers of in front of π n the circe division probem can be principay aso soved, for exampe, for n =, 35, 39, 65. More generay, if the anges π m and π are constructibe, one may consider the combinations n mod m and mod n in anaogous way. π π + with m n. Is Factoriation of Cycotomic Poynomia for n = 9 with 3-Cyces and with Ony Quadratic Radicas Possibe? In case of n= m+ = 9 we find as a first factoriation of the cycotomic poynomia 9 8 i i 9 5 i 9 = = = i 9 3 i 9 + i i 9 3 i i i i 9 i {( )( 4)( 5)( 6)( 7)( 9)( ) ( 6 )( 7 )} { ( )( )( )( )( )( )( )( )( )} = (.) DOI: 0.436/apm Advances in Pure Mathematics