Math 301 Abstract Algebra Final Exam: Wednesday, December 19 2:15 - 4:15 PM. Held in the ordinary class room. 1. 16 puzzle. A B C D A E I M ? ? ? ? E F G H Pi B F J N Pi ? ? ? ? I J K L ========> C G K O ========> ? ? ? ? M N O [] D H L [] ? ? ? [] (a) Write Pi in orbit notation. (b) Write the inverse of Pi in orbit notation. (c) Write Pi^3 in orbit notation. (d) Find ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? [] ------------------------------------------------------------------- Answer A B C D 1 A E I M ? ? ? ? E F G H Pi 5 B F J N Pi ? ? ? ? I J K L ========> 9 C G K O ========> ? ? ? ? M N O [] 13 D H L [] ? ? ? [] (a) (1)(2 5)(3 9)(4 13)(6)(7 10)(8 14)(11)(12 15) (b) The inverse of Pi is Pi. (c) The cube of Pi is Pi. (d) A B C D E F G H I J K L M N O [] ------------------------------------------------------------------- 2. Diaphantine equations. You have a 424 cm^3 scoop and an 742 cm^3 scoop. By adding and subtracting full scoops, what is the smallest positive amount of flour that you can measure. How would you achieve that amount. ------------------------------------------------------------------- ans: 1 424 742 424 --- 1 318 424 318 --- 3 106 318 318 ---- 0 1 1 3 1 1 2 7 0 1 1 4 2 4 - 1 7 = 1 2 4 (106) - 1 7 (106) = 106 2 (424) - 1(742) = 106 Take two of the small scoops minus one of the big scoops. ------------------------------------------------------------------- 3. Puzzle Peg For the following puzzle, assuming it can be solved, where would the last peg be left? # are pegs and o are holes. # # # # # # # # # # # # # # # # # # # o o # # # # # # o o # # # # # # # # # # # # # # # # # # # ------------------------------------------------------------------- answer [# # #] # [# # #] # [# # #] # # [# # #] [# # #] o o [# # #] [# # #] o o [# # #] [# # #] # # [# # #] [# # # ] # [# # # ] # The parity is zero. The game cannot be reduced to a single peg. ------------------------------------------------------------------- 4. The complex numbers. (a) Find e^(ln(2)+3 Pi/2 i) (b) Find the Square root of Cis[30] (c) Graph the 6 sixth-roots of 64. --------------------------------------------- ans. (a) Find e^(ln(2)+3 Pi/2 i) -2i (b) Find the Square root of Cis[30] Cis[15] (c) Graph the 6 sixth-roots of 64. x x x x (2,0) x x 5. Gaussian integers. Find Gaussian Integers a+bi and c+di such that (a+bi)(-21+72 I + (c+di)(6+33 I) = GCD(-21+72I,6+33I) ------------------------------------------------------------------- answer 2 + I 6+33I -21+72I -21+72I -------- 0 GCD(-21+72I,6+33I) = 6+33I 1(6+33I) + 0(-21+72I) = 6+33I ------------------------------------------------------------------- 6. Find c+d Sqrt[3] in Z[Sqrt[3] such that 19 + 37 Sqrt[3] - (c+d Sqrt[3])(3 + Sqrt[3]) has minimal norm. The norm is defined: n(a+b Sqrt[3]) = | a^2 - 3 b^2 | ------------------------------------------------------------------- ans (19+37 Sqrt[3]) (3 - Sqrt[3]) --------------- ------------- = -9 + 46/3 Sqrt[3] (3+ Sqrt[3]) (3 - Sqrt[3]) (19+37 Sqrt[3]) - (3+Sqrt[3])(-9+15 Sqrt[3]) = 1+Sqrt[3] n(1+Sqrt[3]) = 2. ------------------------------------------------------------------- 7. Polya's Counting theorem. How many 8 bead necklaces can be made with 4 blue and 4 green beads? --------------------------------------- answer ()()()()()()()() 8 1 C(8,4) 70 (,,,)(,,,) 2 6 2 2 4 (,)(,)(,)(,) 4 1 C(4,2) 6 (,,,,,,,) 1 3 5 7 0 ()()(,)(,)(,) 4 4 C(4,2) 24 (.)(,)(,)(,) 4 4 C(4,2) 24 ---- 128/16 = 8 ------------------------------------------------------------------- 8. Rational approximations Find integers p and q, both less than 100, such that p/q best approximate Sqrt[3] = 1.73205. ------------------------------------------------------------------- ans 1 100000 173205 100000 ------ 1 73205 100000 73205 ------ 2 26795 73205 53590 ----- 1 19615 26795 19615 ------ 2 7180 19615 14360 ------ 1 5255 7180 5255 ----- 2 1925 5255 3850 ----- 1 1405 1925 1405 ---- 1 520 1405 1040 ---- 265 520 1 1 2 1 2 1 2 1 2 1 1 2 5 7 19 26 71 97 0 1 1 3 4 11 15 41 55 A = {1,2,5/3,7/4,19/11,26/15,71/41,97/55} {1., 2., 1.66667, 1.75, 1.72727, 1.73333, 1.73171, 1.76364} A^2 {1., 4., 2.77778, 3.0625, 2.98347, 3.00444, 2.99881, 3.11041} 9. Groups and Generators. Suppose that {x,y,z} generate a commutative group and satisfy these relations. 2x + 6y + 8 z = 0 4x + 3y -11 z = 0 Express the group in terms of finite cyclic groups and copies of Z. --------------------------------------------- ans a b c 2 6 8 4 3 -11 a b c 2 6 8 0 -9 -27 a+3b+4c b c 2 0 0 0 -9 -27 a+3b+4c b+3c c 2 0 0 0 -9 0 Z2 (+) Z9 (+) Z a+3b+4c = u b+3c = v c = w Solve[{a+3b+4c ==u,b+3c==v,c==w},{a,b,c}] {{a -> u - 3 v + 5 w, b -> v - 3 w, c -> w}} u v w a = ( 1 0 5 ) b = ( 0 1 -3 ) c = ( 0 0 1 u v w 2 a = ( 1 0 5 ) (0 0 10) 6 b = ( 0 1 -3 ) (0 0 -18) 8 c = ( 0 0 1 ) (0 0 8) --------- (0 0 9) u v w 4 a = ( 1 0 5 ) (0 0 20) 3 b = ( 0 1 -3 ) (0 0 -9) -11 c = ( 0 0 1 ) (0 0 -11) ---------- (0 0 0) ----------------------------------------- 10. Subgroups generated: Find the subgroup of S4 generated by. (a) {(1234), (24)} (b) {(1234), (14)} (c) ((124), (132)} (d) {(12),(23),(34)} (e) {(12), (234)} ---------------------------------------------- answer (a) {(1234), (24)} Symmetries of a square 1 2 4 3 (b) {(1234), (14)} S4 (c) ((124), (132)} A4 (d) {(12),(23),(34)} S4 (e) {(12), (234)} S4 ------------------------------------------------------- 11. Solve the cubic: Using radicals, express the solution to the cubic x^3 + 6 x + 7 --------------------------------------------------------- ans f[x_] := x^3 + 6 x + 7 f[z-6/(3z)] = 3 6 -8 + 7 z + z f[z-6/(3z)] = -------------- 3 z z^3 = -8 z^3 = 1 z = -2 z = 1 z = -2 Cis[120] z = Cis[120] z = -2 Cis[240] z = Cis[240] z-2/z = -2+1 = -1 z-2/z = -1 z-2/z = -2Cis[120] +Cis[-120] z-2/z = Cis[120]-2Cis[-120] z-2/z = -2Cis[240] +Cis[-240] z-2/z = Cis[240]-2Cis[-240] x = -1 x = -1 x = -Cos[120] -3 Sin[120] I x = -Cos[120] + 3 Sin[120] I x = -Cos[240] -3 Sin[240] I x = -Cos[240] + 3 Sin[240] I x = -1 x = -Cos[2 Pi/3] - 3 Sin[2 Pi/3] I = 1/2 - 3/2 Sqrt[3] I x = -Cos[4 Pi/3] - 3 Sin[4 Pi/3] I = 1/2 + 3/2 Sqrt[3] I x = -1 x = -Cos[2 Pi/3] + 3 Sin[2 Pi/3] I = 1/2 + 3/2 Sqrt[3] I x = -Cos[4 Pi/3] + 3 Sin[4 Pi/3] I = 1/2 - 3/2 Sqrt[3] I -------------------------------------------------------------- 12. Given the Quartic x^4 + 2 x^2 + 4 x + 2 = 0, What is the cubic polynomial that is necessary to solve so that the polynomial is expressible as the sum of two squares? ------------------------------------------------ ans (x^2 + r)^2 + (s x+t)^2 = x^4 + 2x^2 + 4x + 2 2 2 2 2 4 = r + t + 2 s t x + (2 r + s ) x + x 2 r + s^2 = 2 2 s t = 4 r^2 + t^2 = 2 r = (1/2)(2-s^2) t = 2/s 1/4 (2-s^2)^2 + 4/s^2 - 2 = 0 2 4 6 16 - 4 s - 4 s + s --------------------- = 0 2 4 s 2 4 6 16 - 4 s - 4 s + s = 0 Solve this polynomial for s^2. ----------------------------------------------------- 13. Finite rotations: Suppose one rotates through an angle of 90 degrees about the line {2,2,1} and then 90 degrees about the line {1,2,2}. Where does the point {1,0,0} map to? r1 = {1/Sqrt[2], 2/(3Sqrt[2]),2/(3 Sqrt[2]),1/(3 Sqrt[3])} r1 = {3,2,1,1} r2 = {1/Sqrt[2], 1/(3Sqrt[2], 2/(3Sqrt[2]),2/(3Sqrt[2])} r2 = {3,1,2,2} s1 = 1/15 {3,-2,-1,-1} s2 = 1/15 {3,-1,-2,-2} In[9]:= MatrixForm[r1] Out[9]//MatrixForm= 3 {-2, -1, -1} {2, 1, 1} 3 In[10]:= MatrixForm[s1] Out[10]//MatrixForm= 1 2 1 1 - {--, --, --} 5 15 15 15 2 1 1 1 {-(--), -(--), -(--)} - 15 15 15 5 In[11]:= MatrixForm[r2] Out[11]//MatrixForm= 3 {-1, -2, -2} {1, 2, 2} 3 In[11]:= MatrixForm[r2] Out[11]//MatrixForm= 3 {-1, -2, -2} {1, 2, 2} 3 In[12]:= MatrixForm[s2] Out[12]//MatrixForm= 1 1 1 1 - {--, -, -} 6 18 9 9 1 1 1 1 {-(--), -(-), -(-)} - 18 9 9 6 x[x[x[x[r2,r1],f[{0,1,0,0}]],s1],s2] = Out[14]//MatrixForm= 1 14 2 {-, -(--), -(--)} 0 3 15 15 1 14 2 {-(-), --, --} 3 15 15 0 (1,0,0) is mapped to 1/15 (-5,14,2). --------------------------------------------------------------------