FS23

Task 1

Task 2

$R_9(5^{123})$

Never again the mistake that because in $R_a$ if $a$ not prime, we have to reduce the exponent by $\phi (a)$!

Thus this gives us $\phi (9)=6$!!!

Task 3

e)

Use the property that $F$ is an integral domain and thus the characteristic $k$ is prime.

As a field is a non-trivial ring, we have that $1\ne 0$. Therefore, wlog we can assume that $k\ge 2$. Assume that $k$ is not prime, i.e. there exists $1<p,q<k$ s.t. $pq=k$. Then,$(\underset{p\text{ times}}{\underbrace{1+\cdots +1}})(\underset{q\text{ times}}{\underbrace{1+\cdots +1}})=(\underset{k\text{ times}}{\underbrace{1+\cdots +1}})=0$As $F$ is an integral domain, either $(\underset{p\text{ times}}{\underbrace{1+\cdots +1}})=0$ or $(\underset{q\text{ times}}{\underbrace{1+\cdots +1}})=0$. However, this is a contradiction to the minimality of $k$. Therefore, $k$ is prime.

Task 4

Prenex form

Remember that quantifiers bind harder than operators!! Thus for the prenex form we need parentheses all around.