Abstract Algebra - algebraic structure, and complex numbers

[frogyspond]

Hello,

 

[Limax]

A. Let G be theset of the fifth roots of unity.
1. Use deMoivre’s formula to verify that the fifth roots of unity form a group undercomplex multiplication, showing all work.

1. G is closed under multiplication.
   
   1. Suppose x, y is in the set of G
      
   2. Then \(\displaystyle x^5=y^5=1\).
      
   3. Then \(\displaystyle x^5y^5=(xy)^5=1^5=1\).
      
   4. so that xy is in the set of G.
2. Multiplication on G isassociative because multiplication on C is associative.
   
   1. Note that 1⁵=1
      
   2. so that 1 is in the set of G
      
   3. that for all x is in the set of G
      
   4. we have 1 *x = x*1 = 1
      
   5. So 1 is an identity in G undermultiplication.
3. Suppose x is in the set of G with x⁵= 1
   
   1. Then (x^-1)⁵ =(x⁵)^-1 = 1^-1 = 1
      
   2. so that x^-1 is in the setof G
      
   3. Moreover x * x^-1 = x^-1* x = 1
      
   4. So every element of G has amultiplicative inverse in G.

Thus G is a group under multiplication.


2. Prove that Gis isomorphic to Z₅ under addition by doing the following:
a. State eachstep of the proof.
b. Justify eachof your steps of the proof.

Hint for #2: Consider \(\displaystyle G=\left\{e^{2n\pi i/5}\,:\,n=0,1,2,3,4\right\}\).