Question

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]

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Data

TeamOpponentPart 1Part 2Part 3Total
Chicago AYale A1010020
Chicago BBrown A1010020
Columbia AToronto A0000
Columbia BNYU A0000
Cornell BNorth Carolina A1001020
Duke AMaryland A0000
Florida AStanford A1010020
Georgia Tech ACornell A1010020
Georgia Tech BVirginia A1010020
Harvard ATexas A0000
Houston AFlorida B0000
Indiana ANorthwestern A10101030
Iowa State AIllinois A1010020
Johns Hopkins AMinnesota B100010
McGill ARutgers A1010020
Michigan AImperial A100010
Minnesota AChicago C1010020
Penn APenn State A0000
Purdue AUC Berkeley B0000
Rutgers BVanderbilt A0000
South Carolina AClaremont A100010
UC Berkeley AMIT A1010020
WUSTL AOhio State A0000
Yale BWUSTL B100010