# Algebra: A Computational Introduction by John Scherk

By John Scherk

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Additional resources for Algebra: A Computational Introduction

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Show that under the operations of matrix addition and matrix multiplication, Fp2 is a field with p2 elements. Suggestion: to prove that a non-zero matrix has a multiplicative inverse, use the fact that the congruence x2 − r ≡ 0 (mod p) has no solution. What happens if r is a square in Fp ? 1 Permutations as Mappings In the next chapter, we will begin looking at groups by studying permutation groups. To do this we must first establish the properties of permutations that we shall need there. A permutation of a set X is a rearrangement of the elements of X .

4. • For p prime, 0 < k < p, show that a) ( ) p ≡0 k (mod p) ; b) (x + y)p ≡ xp + y p (mod p) . 5. Prove that a natural number a is divisible by 11 if and only if the alternating sum of its digits is divisible by 11. 6. Show that 10m + n, 0 ≤ n < 10, is divisible by 13 if and only if m + 4n is divisible by 13. 22 CHAPTER 1. CONGRUENCES 7. Prove that a rational number m/n, where m and n are integers, n ̸= 0, has a finite decimal expansion if and only if the prime factors of n are 2 and 5. 8. Let n be a natural number.