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              B   C   D   A             ?   ?   ?   ?
 E   F   G   H      Pi      F   E   H   G     Pi      ?   ?   ?   ?
 I   J   K   L  ========>   K   I   J   L  ========>  ?   ?   ?   ?
 M   N   O  []              M   O   N  []             ?   ?   ?  []

(a)  Write Pi in orbit notation.

(b)  Write the inverse of Pi in orbit notation.

(c)  Write Pi^3 in orbit notation.

(d)  Find   ?


2.  Diaphantine equations.

    You have a 42 cm^3 scoop and an 74 cm^3 scoop.  How

would you measure 2 cm^3.



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   #   #   #
              #   #       #       #   #
              #   #       #       #   #
              #   #   #   #   #   #   #  
              #   #   #   #   #   #   #
            

4.  The complex numbers.

(a)  Find  e^(ln(3)+Pi/2 i)

(b)  Find the Square root of Cis[80]

(c)  Graph the 8 eighth-roots of 64. 



5.  Gaussian integers.

Find Gaussian Integers a+bi and c+di such that

(a+bi)(9+12i) + (c+di)(31+17i) = 1.






6.  Find c+d Sqrt[5] in Z[Sqrt[5] such that  

19 + 37 Sqrt[5] - (c+d Sqrt[5])(3 + Sqrt[5]) has

minimal norm.  The norm is defined: 

n(a+b Sqrt[5])    = | a^2 - 5 b^2 |  


7.  Polya's Counting theorem.

How many 12 bead necklaces can be made with 8 blue

and 4 green beads?


8.  Rational approximations

Find integers p/q which best approximate Sqrt[2].


9.  Groups and Generators.  Suppose that {x,y,z} generate

a commutative group and satisfy these relations. 

               3x + 5y + 8 z = 0
               4x + 3y -12 z = 0

Express the group in terms of finite cyclic groups and copies

of Z.

10.  Subgroups generated: Find the subgroup of S4 generated by.

(a)  {(1234), (13)}

(b)  {(1234), (12)}

(c)  ((123), (234)}

(d)  {(132), (321)}

(e)  {(13), (234)}


11.  Solve the cubic: Using radicals, express the solution 

to the cubic


          x^3  + 18 x + 12 = 0.



12.  Given the Quartic   x^4 + 16 x^2 + 12 x + 8 = 0,

What is the cubic polynomial that is necessary to solve so that 

the polynomial is expressible as the sum of two squares? 

 
13.  Finite rotations: Suppose one rotates through an angle of 

120 degrees about the line {1,2,3} and then 60 degrees about the 

line {3,8,2}.  Where does the point {1,0,0} map to?