Polynomials and fields

Concepts

• Greatest common divisor from Discrete mathematics 10 exercisesThe greatest common divisor gcd(a,b) of two integers a and b, not both zero, is the largest integer that divides both.
• Polynomial ring 10 exercisesThe polynomial ring R[x] over a commutative ring R with identity consists of the finite sums of terms a_ix^i, with coefficients a…
• Polynomial division algorithm 10 exercisesGiven polynomials f and g≠ 0 over a field F, there exist a unique quotient q and a unique remainder r satisfying the equality bel…
• Euclidean domain 10 exercisesAn integral domain D with a norm N (a function from Dsetminus\0\ to the nonnegative integers) allowing division of a by b≠0 with…
• Irreducible element 10 exercisesA nonzero nonunit element p of an integral domain D with no proper factorization: in every product of two factors equal to p, one…
• Unique factorization domain 10 exercisesAn integral domain D in which every nonzero nonunit element a is a product of irreducibles p_i, and this factorization is unique…
• Field of fractions 10 exercisesThe field of fractions Frac(D) of an integral domain D is the field made of the fractions a/b, with a,b∈ D and b≠0, where a/b=c/d…