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

Friday, November 2.  

Main idea: 
            
Key words: norm, Gaussian Integers, e^(Pi i) = -1, roots of unity.

Goal: Learn to relate to the Complex numbers. 


      The complex numbers = {a+bi | a, b in Reals}.  


addition:     a+bi + c+di = (a+c)+(b+d)i

multiplication:  (a+bi)(c+di) = (ac-bd)+(ad+bc)i


      Actually, the process is identical to the process we

already used when we considered Q[Sqrt[2]] = {a+bSqrt[2] | a,b in Q}.

There we treated Sqrt[2] separately until we got two of them

multiplied together and then called it 2.  

      We should be more careful here.  We have defined addition and

multiplication.  But it is not immediately clear that the process is

associative.  One really should check this.

          Check commutativity for addition and multiplication.

          Check associativity for addition and multiplication.

          Check the distributive laws.

          Check that 0+0i is the additive identity.

          Check that 1+0i is the multiplicative identity.


Definition:  | a+bi | = Sqrt[a^2+b^2].


Theorem  |z1 z2| = |z1||z2|

Proof:  |(a+bi)(c+di)| = | (ac-bd)+i(ad+bc)| = Sqrt[(ac-bd)^2 + (ad+bc)^2]

                       = Sqrt[ (ac)^2 + (bd)^2 + (ad)^2 + (bc)^2]

                       = Sqrt[ (a^+b^2)(c^2+d^2)]

                       = |a+bi||c+di|






 
A complex number is said to be in polar form if it is written as

             r(Cos[t]+i Sin[t]).

r is the norm of the complex number and t is called the "argument".

This is often abbreviated as r Cis[t]  where Cis[t] means

Cos[t]+i Sin[t].


Theorem.  Cis[t1] Cis[t2] = r1 r2 Cis[t1+t2].

Proof:  Cis[t1] Cis[t2] = 

       (Cos[t1]+i Sin[t1])(Cos[t2]+i Sin[t2])


 =      (Cos[t1] Cos[t2] - Sin[t1]Sin[t2] +

            i (Cos[t1]Sin[t2] + Sin[t1] Cos[t2])

 =      (Cos[t1+t2]+i Sin[t1+t2]

 =      Cis[t1+t2].


Corollary:   (r Cis[t])^n = r^n Cis[n t].

Problems;

(a) Sqrt[i] = 


(b) CubeRoot[i] = 


(c) Sqrt[1+i] = 


(d) CubeRoot[1] =   


(e) Express Cos[3t] in terms of Cos[t].


Definition:   e^(x+iy) = e^x (Cos[y]+i Sin[y]).


Theorem    e^(z1+z2) = e^(z1) e^(z2)

Proof:    e^(x+iy) e^(u+iv) = e^x Cis[y] e^u Cis[v]

          = e^(x+u) Cis[y+v)

          = e^[(x+iy)+(u+iv)]


Compute 

(a)   e^(1)


(b)   e^(i)


(c)   e^(Pi i).



Definition  Log[z] = a  means e^a = z.

(a) Log[-1]

(b) Log[i]

(c) Log[1+i]

Theorem:  Any polynomial over the Complexes Factors completely.

That is, a polynomial of degree n has n roots (counting multiplicities).

Proof:  This was Gauss' thesis.



Assignment

(a)  Find the |z| and the argument of z when z = 5 + 12 i .


(b)  Plot the five fifth roots of 1.


(c)  Plot the six sixth roots of 1.


(d)  Find the Sqrt[-24 + 70 i]


(e)  Find the Cube Root of -9 + 46 i.


(f)  Find Log[e i]


(g)  Express Cos[4 t in terms of Cos[t].


(h)  Find Gaussian integers x and y such that x(3+5i) + y(2+7i) = 1