One of these algebraic structures whose elements are polynomials with coefficients in some field is always a unique factorization domain. For 10 points each:
[10e] Name these algebraic structures that, unlike groups, are equipped with two binary operations: one akin to addition, the other akin to multiplication.
ANSWER: rings [accept polynomial rings]
[10m] By Hilbert’s basis theorem, if R is a commutative Noetherian (“NUH-ter-ee-an”) ring, then the polynomial ring R[x] (“R-x”) is Noetherian, meaning all of these sets in R[x] are finitely generated. These sets are subrings that are closed under multiplication.
ANSWER: ideals [accept left ideals or right ideals]
[10h] A set F in a polynomial ring is one of these sets if all polynomials in the ideal generated by F can be reduced to zero with respect to F. Computer algebra programs generate these sets with Buchberger’s algorithm.
ANSWER: Gröbner bases (“BAY-sees”) [or Gröbner basis; prompt on standard bases or standard basis or generating sets]
<Other Science>