301 Abstract Algebra
2:00 - 2:50 MWF
Hentzel 432 Carver
hentzel@iastate.edu
Web Page   http://www.math.iastate.edu/hentzel/class.301

Text: A First Course in Abstract Algebra, sixth Edition
by John B. Fraleigh
Course: Chapter 1,2,3,4,5

Wednesday, December 5  

Main Idea:   
Key Words: 
Goal: 


Previous Assignment:

1. Find a dedekind cut which gives the cube root of 2.

upper = { q | q is rational and q^3 >= 2;

lower = { q | q is a rational and q^3 < 2;


2.  Express 3.12121212....  as a rational number.


100x = 312.12121212...
   x =   3.12121212...
-----------------------
 99x = 309

   x = 309/99 = 103/33 = 3.12121212121212121212121212121 

3.  Find a quaternion which rotates through 30 degrees

about the axis vector <1,2,3>.
---------------------------------------------------------------
ans = {Cos[15],0,0,0} + Sin[15]/Sqrt[14] {0,1,2,3}

ans = {Cos[Pi/12],0,0,0} + Sin[Pi/12]/Sqrt[14] {0,1,2,3}

ans = {0.965926, 0.0691723, 0.138345, 0.207517}

------------------------------------------------------------------
If we start with  the quaternion {a,b,c,d} acting on the

three dimensional vector {x,y,z}, then

q2 = {a,b,c,d}*{0,x,y,z}*{a,b,c,d}^(-1) = 

{a,b,c,d}*{0,x,y,z}*{a,-b,-c,-d}/Sqrt[a^2+b^2+c^2+d^2] = 


1/(a^2+b^2+c^2+d^2) 

         {0,

            2    2    2    2
          (a  + b  - c  - d ) x + (2 b c - 2 a d) y + (2 a c + 2 b d) z,

       
                                2    2    2    2
          (2 b c + 2 a d) x + (a  - b  + c  - d ) y + (-2 a b + 2 c d) z,

                                                     2    2    2    2
          (-2 a c + 2 b d) x + (2 a b + 2 c d) y + (a  - b  - c  + d ) z}


The starting point is q1 = {x,y,z}.

The point on the line {b,c,d} about which the point rotates is

      (b x + c y + d z)
p0 = ------------------ (b,c,d)
       (b^2+c^2+d^2)


              q2                  q1
               \                  /
                \                /
                 \              /
                  \            /
                   \          /
                    \ theta  /
                     \      /
                      \    /
                       \  /
                        \/     
                         p0

                                     2    2    2    2
             (q1-p0).(q2-p0)        a  - b  - c  - d
Cos[theta] = ----------------  =   -------------------
             |q1-p0| |q2-p0|         2    2    2    2
                                    a  + b  + c  + d

                  2 a ^2
          =  ---------------  - 1
             a^2+b^2+c^2+d^2 


                     a^2
          =   2 --------------- -1
                a^2+b^2+c^2+d^2 

                     a
Notice that   ---------------   is the first component of  
              a^2+b^2+c^2+d^2 

the unit quaternion made from (a,b,c,d).  So if we call this

Cos[A], then the Cos[theta] = 2 Cos[A]^2 - 1  = Cos[2 A].

Thus if you want to build the quaternion that rotates through

an angle theta about an axis along  V, use

                                      V
{Cos[theta/2],0,0,0} + Sin[theta/2] -----.
                                     |V|



The relation between the quaternions and the octonions.

This is the multiplication of Octonions.  

     |a  B|  |e  F|   = | ae+B.G,        aF + h B - CXG |
     |C  d|  |G  h|     | eC+dG +BXF           C.F + dh |








The norm of an octonion is the "determinant" = 

n| a  B |  = ad-B.C
 | C  d |

and a major property about this norm is n[XY] = n[X] n[Y].

This algebra is the largest algebra which has such a 

(non-degenerate) norm.


There is a subring of these octonion matrices that behaves

like the quaternions.



     |a -B|  |e -F|   = | ae-B.F,       -aF - e B - BXF |
     |B  a|  |F  e|     | eB+aF +BXF          -B.F + ae |


The norm of a quaternion is the "determinant" = a^2 +B.B

So if we write out the matrix form: 

 a+bi+cj+dk = | a         {-b,-c,-d}|
              | { b, c, d}     a    |

n(a+bi+cj+dk) = a^2 -{-b,-c,-d}.{b,c,d} = a^2 + b^2 + c^2 + d^2



      Now we have to go the the realm of Hocus Pocus, or so it

might seem.  The norm of an Octonion is 


n| a,    {b,c,d}|  = ah -be -cf -dg.
 |{e,f,g}      h|

This should be, if things go as promised a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2.

In the algebra consisting of |a  B|  we have to decide where the 
                             |C  d|

elements come from.  If they come from the complex numbers, then

we have the complex-octonions.  If they come from the real numbers,

then we have real octonions.  But there is another way to get

real octonions.  Consider "skew symmetric" matrices over the

quaternions.

         ___         ___          ___             ___  ____
  |a     -B |  | c   -D  |   | ac -B.D          -a D  -c B  - BxD  |
  |      ___|  |     ___ | = |     ___   ___ ___         ___  ____ |
  |B      a |  | D    c  |   | cB + a D + B x D        -B.D  + a c |


Counting the real and complex parts separately, this is also

an eight dimensional algebra.  The typical element is


     | a+bi              -{c-di,e-fi,g-hi}|
     | {c+di,e+fi,g+hi}     a-bi          |


The norm of the above is a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2.



Assignment: (1) rotate 60 degrees about the line { 1, 1, 1}.

            (2) rotate 60 degrees about the line {-1, 1,-1}.

            (3) rotate 60 degrees about the line { 1,-1,-1}.


(a) Give the final positions of the points {1,0,0},{0,1,0},{0,0,1}.

(b) Give the rotation and the axis to return everything back the


r1 =  {3,  1, 1,  1}

r2 =  {3, -1, 1, -1}

r3 =  {3,  1,-1, -1}





r2xr2xr1 = {32, 24, 8, -8}

Cos[1/2 theta] = 32/Sqrt[1728] =  4/(3 Sqrt[3])

theta = 79.3281 degrees.

The inverse is to rotate -79.3281 about the axis { 3,1,-1}


{1,0,0} --->  1/27 {23,2,14};

{0,1,0} ----> 1/27 {14,7,22}; 

{0,0,1} ----> 1/27 {2, 26, 7};