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Elements Of Modern Algebra

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.6: Congruence Classes

Problem 18E

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Let p be a prime number. The polynomial p(x) = x²+x+1 is irreducible over Zp if and
and only if either p = 5 mod 12 or p = 11 mod 12.

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30. Prove statement of Theorem : for all integers .
Prove that if p and q are distinct primes, then there exist integers m and n such that pm+qn=1.
Find all monic irreducible polynomials of degree 2 over Z3.
54. Let be a prime integer. Prove Fermat's Little Theorem: For any positive integer,. (Hint: Use induction on, with held fixed.)
Label each of the following statements as either true or false. a is congruent to b modulo n if and only if a and b yield the same remainder when each is divided by n.
a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is divisible by 11 if and only if 11 divides a0-a1+a2-+(1)nan, when z is written in the form as described in the previous problem. a. Prove that 10n1(mod9) for every positive integer n. b. Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. (Hint: Any integer can be expressed in the form an10n+an110n1++a110+a0 where each ai is one of the digits 0,1,...,9.)
Let and be positive integers. If and is the least common multiple of and , prove that . Note that it follows that the least common multiple of two positive relatively prime integers is their product.

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