Question

. Provide an Example for each of the followings. If there is Not any example explain why. A finite field : • A commutative ring with zero divisors : An integral domain that is not a field : A non-abelian cyclic group : A cyclic group of order 36: • A non-abelian group of order 10 : . .

Answer 1

(1) The simplest finite field is $F_2=\{0,1\}$ w.r.t. the operations $+_2$ and  $\times_2$.

(2) A simple example is $\mathbb Z_6=\{0,1,2,3,4,5\}$ where

$2\times_6 3=0$

hence 2 and 3 are zero divisors.

(3) An example is set of integers $\mathbb Z=\{.......-3,-2,-1,0,1,2,3......\}$

which is an integral domai but not a field as multiplicative inverse doesn't exist for elements like 2,3,4,..etc.

(4) Such example doesn't exist because "Every cyclic group is abelian".

(5) An example is $\mathbb Z_{36}=\{0,1,2,3,.....,35\}$ which is cyclic as 1 is generator.

(6) An example is group of symmetries $D_{10}.$