Chapter 20

We will see later that the multiplicative group of nonzero elements of a finite field is cyclic. Illustrate this by finding a genentor for this group for the given finite field. $$ Z_{7} $$

We will see later that the multiplicative group of nonzero elements of a finite field is cyclic. Illustrate this by finding a genentor for this group for the given finite field. $$ \mathbb{Z}_{11} $$

We will see later that the multiplicative group of nonzero elements of a finite field is cyclic. Illustrate this by finding a genentor for this group for the given finite field. $$ \mathrm{Z}_{17} $$

We will see later that the multiplicative group of nonzero elements of a finite field is cyclic. Illustrate this by finding a genentor for this group for the given finite field. $$ \text { Using Fermat's theorem, find the remainder of } 3^{47} \text { when it is divided by } 23 . $$

We will see later that the multiplicative group of nonzero elements of a finite field is cyclic. Illustrate this by finding a genentor for this group for the given finite field. $$ \text { Use Fermat's theorem to find the remainder of } 37^{49} \text { when it is divided by } 7 . $$

$$ \text { In Exercises } 11 \text { through } 18 \text {, describe all solutions of the given congruence, as we did in Examples } 20.14 \text { and 20.15. } $$ $$ 2 x=6(\bmod 4) $$

$$ \text { In Exercises } 11 \text { through } 18 \text {, describe all solutions of the given congruence, as we did in Examples } 20.14 \text { and } 20.15 \text {. } $$ $$ 36 x \equiv 15(\bmod 24) $$

$$ \text { In Exercises } 11 \text { through } 18 \text {, describe all solutions of the given congruence, as we did in Examples } 20.14 \text { and } 20.15 \text {. } $$ $$ 45 x \equiv 15(\bmod 24) $$

Using Exercise 28 below, find the remainder of \(24 !\) modulo \(29 .\)

Mark each of the following true or false. _____a. \(a^{p-1} \equiv 1(\bmod p)\) for all integers \(a\) and primes \(p\). _____b. \(a^{p-1}=1\) (mod \(\left.p\right)\) for all integers \(a\) such that \(a \neq 0(\bmod p)\) for a prime \(p\). _____c. \(\varphi(n) \leq n\) for all \(n \in \mathbf{Z}^{+}\). _____d. \(\psi(n) \leq n-1\) for all \(n \in 2^{+}\). i. Every congruence \(a x=b\) (mod \(p\) ), where \(p\) is a prime, has a solution. j. Let \(d\) be the ged of positive integers \(a\) and \(m\). If \(d\) divides \(b\), then the congruence \(a x \equiv b(\bmod m)\) has exactly \(d\) incongruent solutions. _____e. The units in \(Z_{n}\) are the positive integers less than \(n\) and relatively prime to \(n\). _____f. The product of two units in \(Z_{n}\) is always a unit. _____g. The product of two nonunits in \(\mathrm{R}_{m}\) may be a unit. _____h. The product of a unit and a nonunit in \(Z_{n}\) is never a unit.

Give the group multiplication table for the multiplicative group of units in \(Z_{12}\). To which group of order 4 is it isomorphic?

Show that 1 and \(p-1\) are the only elements of the field \(Z_{p}\) that are their own multiplicative inverse.