Kunze Linear Algebra · Problem 4.5.5

Polynomials

Mathematics — Methods · congruence-modulo-polynomial-equivalence

Problem

Prove, for any non-zero polynomial pp, that congruence modulo pp is an equivalence relation. ((a)) It is reflexive: f≡f mod pf \equiv f \bmod p. ((b)) It is symmetric: if f≡g mod pf \equiv g \bmod p, then g≡f mod pg \equiv f \bmod p. ((c)) It is transitive: if f≡g mod pf \equiv g \bmod p and g≡h mod pg \equiv h \bmod p, then f≡h mod pf \equiv h \bmod p. (Definition: If ff, gg, and pp are polynomials over the field FF with p≠0p \neq 0, we say that ff is congruent to gg modulo pp if (f−g)(f-g) is divisible by pp. If ff is congruent to gg modulo pp, we write f≡g mod pf \equiv g \bmod p.)

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