Answer to Question #193280 in Abstract Algebra for Neena peter

Question

1 Let f be a non trivial homomorphism from Z10 to Z15.Then which of the following holds?

A) im f is of order 10

B) ker f is of order 5

C) ker f is of order 2

D) f is a one to one map

2.the number of zeros of z⁵+3z²+1 in |z|<1,counted with multiplicity is

A)0 B) 1 C)2 D)3

How to solve this problems.

Answer 1

2)

$|z^5+1|\le1+1=2<3=3|z^2|=|3z^2|$

$3z^2$ has two zeros.

By Rouché's theorem $z^5+3z^2+1$ has also two zeros.

Answer: C) 2.

1)

Kernel of f consists of integers in $Z_{10}$ and $Z_{15}$ which are divisible by $5=gcd(|Z_{10},Z_{15})$

ker f$=\{5,10\}$

Answer: C) ker f is of order 2