مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 On Projective 3-Space Over Galo is Field A. SH. Al-Muk htar Departme nt of Mathematics, College of Education I bn-Al-Haitham ,Unive rsity of Baghdad Received in : 11 May 2011 Accepte d in :16 June 2011 Abstract The p urp ose of this p aper is to give the definition of p rojective 3-sp ace PG(3,q) over Galois field GF(q), q = p m for some prime number p and some integer m. Also, t he definition of the p lane in PG(3,q) is given and state the principle of duality . M oreover some theorems in PG(3,q) are p roved. Keywords: p lane, duality , Galois field. 1- Introduction, [1,2] A p rojective 3 – sp ace PG(3,K) over a field K is a 3 – dimensional p rojective sp ace which consist s of p oints, lines and planes with the incidence relation between them. The projective 3 – sp ace satisfies the following axioms: A. Any two dist inct p oints are contained in a unique line. B. Any three distinct non-collinear p oints, also any line and p oint not on the line are contained in a unique p lane. C. Any two dist inct coplanar lines intersect in a unique p oint. D. Any line not on a given p lane intersects t he plane in a unique point. E. Any two dist inct p lanes intersect in a unique line. A p rojective sp ace PG(3,q) over Galois field GF(q), q = p m , for some p rime number p and some integer m, is a 3 – dimensional p rojective sp ace. Any p oint in PG(3,q) has the form of a quadrable (x1, x2, x3, x4), where x1, x2, x3, x4 are elements in GF(q) with t he excep tion of the quadrable consisting of four zero elements. Two quadrables (x1, x2, x3, x4) and (y1, y 2, y 3, y 4) represent t he same point if there exists  in GF(q) \ {0} such that (x1, x2, x3, x4) =  (y 1, y 2, y 3, y 4), this is denoted by (x1, x2, x3, x4)  (y 1, y 2, y 3, y 4). Similarly , any p lane in PG(3,q) has the form of a quadrable [x1, x2, x3, x4], where x1, x2, x3, x4 are distinct elements in GF(q) with the excep tion of the quadrable consisting of four zero elements. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 Two quadrables [x1, x2, x3, x4] and [y1, y 2, y 3, y 4] represent t he same plane if there exists  in GF(q) \ {0} such that [x1, x2, x3, x4] =  [y 1, y 2, y 3, y 4], this is denoted by [x1, x2, x3, x4]  [y 1, y 2, y 3, y 4].. Also a p oint P(x1, x2, x3, x4) is incident with the p lane  [a1, a2, a3, a4] iff a1 x1 + a2 x2 + a3 x3 + a4 x4 = 0. De fini tion 1.1: [2] A p lane  in PG(3,q) is the set of all p oints P(x1, x2, x3, x4) satisfy ing a linear equation u1 x1 + u2 x2 + u3 x3 + u4 x4 = 0. This p lane is denoted by  [u1, u2, u3, u4]. It should be noted that if one takes another representation of P, say ( x1,  x2,  x3,  x4), then since u1  x1 + u2  x2 + u3  x3 + u4  x4 =  (u1 x1 + u2 x2 + u3 x3 + u4 x4), the definition of a plane is independent of the choice of rep resentations of p oints on it. 2- Principle of Duality De fini tion 2.1: [3] For any S = PG(n,K), there is a dual sp ace S*, whose p oints and p rimes (subsp aces of dimensions (n – 1)) are resp ectively the p rimes and p oints of S. For any theorem true in S, there is an equivalent theorem true in S*. In p articular, if T is a theorem in S st ated in terms of p oints, p rimes and incidence, the same theorem is true in S* and gives a dual theorem T* in S by interchanging "p oint" and "p rime" whenever they occur. In PG(3,K) p oint and p lane are dual, where as t he dual of a line is a line. The orem 2.2: The p oints of PG(3,q) have unique forms which are (1,0,0,0), (x,1,0,0), (x, y ,1,0), (x, y , z,1) for all x, y , z in GF(q). Proof : Let P(x1, x2, x3, x4); x1; x2, x3, x4GF(q) be any p oint in PG(3,q), then either x40 or x4=0. If x4  0, then P(x1, x2, x3, x4)  31 2 4 4 4 ( , , ,1) xx x x x x , where 1 4  x x x , 2 4  x y x , 3 4  x z x . If x4 = 0, then either x3  0 or x3 = 0. If x3  0, then P(x1, x2, x3, 0)  1 2 3 3 ( , ,1, 0) x x x x , where 1 3  x x x , 2 3  x y x . If x3 = 0, then either x2  0 or x2 = 0. If x2  0, then P(x1, x2, 0, 0)  1 2 ( ,1, 0, 0) x x = P(x, 1, 0, 0), where 1 2  x x x . مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 If x2 = 0, then x1  0 and P(x1, 0, 0, 0)  1 1 ( , 0, 0, 0) x x = P(1, 0, 0, 0). Similar ly, one can prove the dual of theorem 1. The orem 2.3: The p lanes of PG(3,q) have unique forms which are [1,0,0,0], [x,1,0,0], [x, y ,1,0], [x, y , z,1] for all x, y , z in GF(q). Th e ore m 2.4: [1 ] Every line in PG(3,q) contains exactly q + 1 p oints. The orem 2.5: [1] Every p oint in PG(3,q) is on exactly q + 1 lines. The orem 2.6: [1] Every p lane in PG(3,q) contains exactly q 2 + q + 1 p oints (lines). The orem 2.7: [1] Every p oint in PG(3,q) is on exactly q 2 + q + 1 p lanes. The orem 2.8: There exist q 3 + q 2 + q + 1 p oints in PG(3,q). Proof : From theorem 1, the p oints of PG(3,9) have unique forms which are (1,0,0,0), (x,1,0,0), (x, y ,1,0), (x, y , z,1) for all x, y , z in GF(q). It is clear that t here exists one p oint of t he form (1,0,0,0). There exist q p oints of the form (x,1,0,0). There exist q 2 p oints of the form (x, y ,1,0). There exist q 3 p oints of the form (x, y , z,1). S imilarly, one can prove the dual of the orem 2.8. The orem 2.9: There exist q 3 + q 2 + q + 1 p lanes in PG(3,q). The orem 2.10: Any two p lanes in PG(3,q) intersect in exactly q + 1 p oints. Proof : By axiom E, since any two p lanes intersect in a unique line and each line in PG(3,q) contains exactly q + 1 p oints, then any two p lanes intersect in exactly q + 1 p oints. The orem 2.11: مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 Any line in PG(3,q) is on exactly q +1 p lanes. Proof : Let l be any line in PG(3,q) and m be another line in PG(3,q) not coplanar with l . m contains exactly q + 1 p oints. By axiom B, l determines a unique plane with any p oint of m. Hence there exist q + 1 p lanes through l . If there exists another p lane through l , then this p lane intersects m in another p oint which is a contradiction. Hence l is on exactly q + 1 p lanes. The orem 2.12: Any two p oints in PG(3,q) are on exactly q + 1 p lanes. Proof : Since any two p oints determine a unique line and by theorem 10, then every line is on exactly q + 1 p lanes. The orem 2.13: There exist (q 2 + 1) (q 2 + q + 1) lines in PG(3,q). Proof : In PG(3,q), there exist q 3 + q 2 + q + 1 p lanes, and each p lane contains exactly q 2 + q + 1 lines, then the numbers of lines is equal to (q 3 + q 2 + q + 1)( q 2 + q + 1), but each line is on q + 1 p lanes, then there exist exactly 3 2 2 2 2( 1)( 1) ( 1)( 1) ( 1)           q q q q q q q q q lines in PG(3,q). Now, some theorems on p rojective 3-sp ace PG(3,q) can be p roved. The orem 2.14: Four distinct p oints A(x1, x2, x3, x4), B(y 1, y 2, y 3, y 4), C(z1, z2, z3, z4), and D(w1, w2, w3, w4) are coplanar iff 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 0   x x x x y y y y z z z z w w w w Proof : Let  [u1, u2, u3, u4] be a p lane containing the points A, B, C, D, then x1 u1 + x2 u2 + x3 u3 + x4 u4 = 0 y 1 u1 + y 2 u2 + y 3 u3 + y 4 u4 = 0 z1 u1 + z 2 u2 + z 3 u3 + z 4 u4 = 0 w1 u1 + w2 u2 + w3 u3 + w4 u4 = 0 It is known from the linear algebra that this sy st em of equations have non zero solutions for u1,u2, u3,u4 iff  = 0. Thus the necessary and sufficient conditions for four p oints to be coplanar that  = 0. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 Corollary 2.15: If four distinct p oints in PG(3,q) A(x1, x2, x3, x4), B(y 1, y 2, y 3, y 4), C(z1, z2, z3, z4), and D(w1, w2, w3, w4) are collinear, then  = 0. This follows from theorem 2.14 and the incidence of these p oints on a line of some p lane. From the principle of duality , one can prove: The orem 2.16: Four distinct p lanes in PG(3,q) A[x1, x2, x3, x4], B[y 1, y 2, y 3, y 4], C[z1, z2, z3, z4], and D[w1, w2, w3, w4] are concurrent (intersecting in one p oint) iff 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 0   x x x x y y y y z z z z w w w w The orem 2.17: The equation of the p lane determined by three distinct p oints A(y 1, y 2, y 3, y 4), B(z1, z2, z3, z4), and C(w1, w2, w3, w4) is 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 2 3 4 3 1 4 1 2 4 3 2 1 2 3 4 1 3 1 4 2 1 2 4 3 3 2 1 4 2 3 4 3 1 4 1 2 4 3 2 1 0      x x x x y y y y z z z z w w w w y y y y y y y y y y y y z z z x z z z x z z z x z z z x w w w w w w w w w w w w where (x1, x2, x3, x4) be any variable point on t he plane, and it’s coordinates are: 2 3 4 3 1 4 1 2 4 3 2 1 2 3 4 3 1 4 1 2 4 3 2 1 2 3 4 3 1 4 1 2 4 3 2 1 , , ,           y y y y y y y y y y y y z z z z z z z z z z z z w w w w w w w w w w w w Similarly , one can prove the dual of this theorem. The orem 2.18: The equation of the p oint determined by three distinct p lanes (non-collinear) in PG(3,q) a[y1, y 2, y 3, y 4], b[z1, z2, z3, z4], and c[w1, w2, w3, w4] is 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4  x x x x y y y y z z z z w w w w مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 2 3 4 3 1 4 1 2 4 3 2 1 2 3 4 1 3 1 4 2 1 2 4 3 3 2 1 4 2 3 4 3 1 4 1 2 4 3 2 1 0    y y y y y y y y y y y y z z z x z z z x z z z x z z z x w w w w w w w w w w w w where [x1, x2, x3, x4] be any variable plane passing through the point, and it’s coordinates are: 2 3 4 3 1 4 1 2 4 3 2 1 2 3 4 3 1 4 1 2 4 3 2 1 2 3 4 3 1 4 1 2 4 3 2 1 , , ,           y y y y y y y y y y y y z z z z z z z z z z z z w w w w w w w w w w w w Notati on 2.19: If v is the vector with comp onents (a1, a2, a3, a4), then the sy mbol P(v) means that the coordinates of t he point P are (a1, a2, a3, a4) in a p rojective 3–sp ace S = PG(3,K). De fini tion 2.20:[3] The p oints Pi(vi), with i = 1, …, m are linearly dependent or indep endent according as the vectors vi are linearly dependent or indep endent. De fini tion 2.21:[3] If the p oints P1, P2, , Pm are linearly dependent, then at least one of the ci’s of the equation 1 ( ) 0    m i i i i c v is not equal to zero, say c1, then P1 = 1 1 c ( c2 P2 + c3 P3 +  + cm Pm ). The point P1 is then said to be a linear combination of the p oints P2, P3, , Pm . This definition may be dualized by replacing the word "p oint" by the word "p lane", and the geometric meaning of linear dependence of p oints or p lanes may now be given. The orem 2.22: Two p oints (p lanes) in PG(3,q) are linearly dependent iff they coincide. Proof : Let P and Q be any two p oints. If P and Q are linearly dependent, then there exist c1 and c2 such that (c1, c2)  (0,0), c1 P + c2 Q = . If c1 = 0, then c2 Q = . This imp lies c2 = 0, since Q  (0,0,0). Then c1  0 and similarly c2  0, 2 1 c Q c   . This means that P and Q coincide. If P and Q are coincide, then there exist c1 0, c2  0 s.t . c1 P = c2 Q. Hence, c1 P  c2 Q =  and thus P and Q are linearly dependent. The orem 2.23: Four p oints in PG(3,q) are linearly dependent iff they are coplanar. Proof : مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 Let A(x1, x2, x3, x4), B(y 1, y 2, y 3, y 4), C(z1, z2, z3, z4), and D(w1, w2, w3, w4) be any four p oints in S. If A, B, C, D are linearly dependent, then there exist c1, c2, c3 and c4 in K such that (c1, c2, c3, c4)  (0,0,0,0) and c1 A+ c2 B+ c3 C + c4 D =  c1 A + c2 B + c3 C + c4 D = c1 (x1, x2, x3, x4) + c2 (y 1, y 2, y 3, y 4) + c3 (z1, z2, z3, z4) + c4 (w1, w2, w3, w4) = (0,0,0,0) c1 x1 + c2 y 1 + c3 z1 + c4 w1 = 0 c1 x2 + c2 y 2 + c3 z2 + c4 w2 = 0 c1 x3 + c2 y 3 + c3 z3 + c4 w3 = 0 c1 x4 + c2 y 4 + c3 z4 + c4 w4 = 0 (1) This sy st em has non zero solutions for c1, c2, c3, c4 iff 1 2 3 41 1 1 1 1 2 3 42 2 2 2 1 2 3 43 3 3 3 1 2 3 44 4 4 4 0    x x x xx y z w y y y yx y z w z z z zx y z w w w w wx y z w by theorem 2.14 the p oints A, B, C, D are coplanar. Conversely , if the p oints A, B, C, D are coplanar, then 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 0   x x x x y y y y z z z z w w w w , then 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 0 x y z w x y z w x y z w x y z w . So the sy st em (1) of equations has non zero solutions for c1, c2, c3, c4. Thus A, B, C, D are linearly dependent. The orem 2.24: Any five p oints (p lanes) in PG(3,q) in S are linearly dependent. Proof : Let A(a1, a2, a3, a4), B(b1, b2, b3, b4), C(c1, c2, c3, c4), D(d1, d2, d3, d4) and E(e1, e2, e3, e4) be any five p oints in S. Let a A + b B + c C + d D + e E =  a (a1,a2,a3,a4) + b (b1,b2,b3,b4) + c (c1,c2,c3,c4) + d (d1,d2,d3,d4) + e (e1,e2,e3,e4) =  a a1 + b b1 + c c1 + d d1 + e e1 = 0 a a2 + b b2 + c c2 + d d2 + e e2 = 0 a a3 + b b3 + c c3 + d d3 + e e3 = 0 a a4 + b b4 + c c4 + d d4 + e e4 = 0 This sy st em of 4 linear homogeneous equations in 5 unknowns a, b, c, d, e has non trivial solutions since 4 < 5. Then A, B, C, D, E are linearly dependent. The orem 2.25: مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 In PG(3,q) if P1, P2, , Pm are linearly indep endent p oints while P1, P2, , Pm + 1 are linearly dependent, then the coordinates of the p oints may be chosen so that P1 + P2 +  + Pm = Pm + 1. Proof : Since the p oints P1, P2, , Pm + 1 are linearly dependent, constants c1, c2, , cm + 1  0, 0, , 0 exist such that c1 P1(v1) + c2 P2(v2) +  + cm Pm (vm ) + cm + 1 P m + 1(v m + 1) = . Now, cm + 1  0, for otherwise the p oints P1, P2, , Pm would be dependent contrary to hy p othesis. The equation may , therefore, be solved for Pm + 1 giving Pm + 1 = m 1 1 c   [ c1 P1(v1) +  + cm Pm (vm ) ] = k1 P1(v1) +  + km Pm (vm ) = P1(k1 v1) +  + Pm (km vm ) where 1   i i m c k c , i = 1, , m or dropp ing the sy mbols ki vi , Pm + 1=P1+ P2++Pm . The orem 2.26: In PG(3,q) a point D is on the plane determined by three distinct p oints A, B, C iff D is a linear combination of A, B, C. Proof : If D is on t he plane determined by three distinct p oints, then A, B, C, D are cop lanar. By theorem (5), they are linearly dependent, there exist constants a, b, c, d such that not all of them are zero and a A + b B + c C + d D = . If d = 0, then a A + b B + c C = , which implies that a = b = c = 0, since A, B, C are linearly indep endent, which is a contradiction. Since any three noncollinear p oints in the p lane are linearly indep endent, [3]. So d  0, and then D = ( ) a d A + ( ) b d B + ( ) c d C Thus D is a linear combination of A, B, C. Sup p ose D is a linear combination of A, B, C, then there exist constants c1, c2, c3 not all of them are zero such that: D = c1 A + c2 B + c3 C, which implies c1 A + c2 B + c3 C + (–1) D = , then it follows that A, B, C, D are linearly dependent. By theorem (5), the p oints A, B, C, D are coplanar. Re ferences 1. Al-M ukhtar, A.Sh. (2008) Complete Arcs and Surfaces in three Dimensional Projective Sp ace Over Galois Field, Ph.D. Thesis, University of Technology , Iraq. 2. Kirdar,M .S. and Al-M ukhtar, A.Sh. (2009) Engineering and Technology Journal, On Projective 3-Sp ace, Vol.27(8): مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 3. Hirschfeld, J. W. P. (1998) Projective Geometries Over Finite Fields, Second Edition, Oxford University Press. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 حول الفضاء الثالثي االسقاطي حول حقل كالوا آمال شهاب المختار جامعة بغداد، ابن الهیثم -كلیة التربیة، قسم الریاضیات 2011 حزیران 16: قبل البحث في 2011 آیار 11:استلم البحث في الخالصة ـــل كــــالوا PG(3,q)الغـــرض مــــن هـــذا البحــــث هـــو إعطــــاء تعریـــف الفــــضاء الثالثـــي االســــقاطي ، GF(q) فـــي حقـ q = p m ونص PG(3,q)كذلك تقدیم تعریف المستوي في . عدد صحیحm عدد أولي و p،اذ ان m و p ، لبعض قیم .PG(3,q)مبدأ الثنائیة وبرهنت بعض المبرهنات في . مستوي ، مبدأ الثنائیة ، حقل كالوا:الكلمات المفتاحیة