Question #193280

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.


1
Expert's answer
2021-05-17T16:52:45-0400

2)

z5+11+1=2<3=3z2=3z2|z^5+1|\le1+1=2<3=3|z^2|=|3z^2|

3z23z^2 has two zeros.

By Rouché's theorem z5+3z2+1z^5+3z^2+1 has also two zeros.

Answer: C) 2.


1)

Kernel of f consists of integers in Z10Z_{10} and Z15Z_{15} which are divisible by 5=gcd(Z10,Z15)5=gcd(|Z_{10},Z_{15})

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

Answer: C) ker f is of order 2


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