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Chapter 14: Problem 18

For \(p\) a prime determine all elements \(a \in \mathbf{Z}_{p}\) where \(a^{2}=a\).

Short Answer

Expert verified
The elements that satisfy \(a^{2}=a\) for \(a\) in \(\mathbf{Z}_{p}\) for any prime \(p\) are \(a=0\) and \(a=1\).

Step by step solution

01

Define the Problem

Given: A prime number \(p\), an integer \(a\) such that \(a^{2}=a\).\n\nTo find: All such \(a\)'s in \(\mathbf{Z}_{p}\).
02

Rearrange the Equation

Rearrange the given equation \(a^{2}=a\) to form \(a^{2}-a=0\). This equation can be factorised as \(a(a-1)=0\).
03

Applying Modulo Operation

From \(a(a-1) = 0\), we have two solutions for \(a\), either \(a=0\) or \(a=1\). This holds true for all \(\mathbf{Z}_{p}\).
04

Conclusion

By solving for all \(a\) in \(\mathbf{Z}_{p}\), we find that only \(a=0\) and \(a=1\) satisfy the equation \(a^{2}=a\).

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Most popular questions from this chapter

Let \(R=M_{2}(\mathbf{Z})\) and let \(S\) be the subset of \(R\) where $$ S=\left\\{\left[\begin{array}{cc} x & x-y \\ x-y & y \end{array}\right] \mid x, y \in \mathbf{Z}\right\\} . $$ Prove that \(S\) is a subring of \(R\).

Give an example of a ring with eight elements. How about one with 16 elements? Generalize.

a) In how many ways can one select two positive integers \(m, n\), not necessarily distinct, so that \(1 \leq m, n \leq 100\) and the last digit of \(7^{m}+3^{n}\) is \(8 ?\) b) Answer part (a) for the case where \(1 \leq m, n \leq 125\). c) If one randomly selects \(m, n\) [as in part (a)], what is the probability that 2 is now the last digit of \(7^{m}+3^{n}\) ?

Find the multiplicative inverse of each element in \(\mathbf{Z}_{11}, \mathbf{Z}_{13}\), and \(\mathbf{Z}_{17}\).

Let \(R\) be a ring with ideals \(A\) and \(B\). Define \(A+B=\\{a+b \mid a \in A, b \in B\\}\). Prove that \(A+B\) is an ideal of \(R\). (For any ring \(R\), the ideals of \(R\) form a poset under set inclusion. If \(A\) and \(B\) are ideals of \(R\), with glb \(\\{A, B\\}=A \cap B\) and lub \(\\{A, B\\}=A+B\), the poset is a lattice.)
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