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Let p be a prime.a. Determine the number of irreducible

Chapter 17, Problem 18E

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QUESTION:

Problem 18E

Let p be a prime.

a. Determine the number of irreducible polynomials over Zp of the form x2 + ax + b.

b. Determine the number of irreducible quadratic polynomials over Zp.

Questions & Answers

QUESTION:

Problem 18E

Let p be a prime.

a. Determine the number of irreducible polynomials over Zp of the form x2 + ax + b.

b. Determine the number of irreducible quadratic polynomials over Zp.

ANSWER:

Step 1 of 3

Recall that the number of elements in  are .

Note that  and  can be chosen in p ways each. Hence, these can be chosen in  ways.

a)

Suppose is reducible. Then it can be factorized in the following way:

                                                         

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