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]
<Other Science>
Summary
| 2023 ACF Nationals | 04/22/2023 | Y | 24 | 11.25 | 63% | 42% | 8% |
Data
| Minnesota A | Chicago C | 10 | 10 | 0 | 20 |
| Cornell B | North Carolina A | 10 | 0 | 10 | 20 |
| Chicago A | Yale A | 10 | 10 | 0 | 20 |
| Houston A | Florida B | 0 | 0 | 0 | 0 |
| Michigan A | Imperial A | 10 | 0 | 0 | 10 |
| Johns Hopkins A | Minnesota B | 10 | 0 | 0 | 10 |
| UC Berkeley A | MIT A | 10 | 10 | 0 | 20 |
| McGill A | Rutgers A | 10 | 10 | 0 | 20 |
| WUSTL A | Ohio State A | 0 | 0 | 0 | 0 |
| Penn A | Penn State A | 0 | 0 | 0 | 0 |
| South Carolina A | Claremont A | 10 | 0 | 0 | 10 |
| Columbia A | Toronto A | 0 | 0 | 0 | 0 |
| Georgia Tech B | Virginia A | 10 | 10 | 0 | 20 |
| Harvard A | Texas A | 0 | 0 | 0 | 0 |
| Iowa State A | Illinois A | 10 | 10 | 0 | 20 |
| Columbia B | NYU A | 0 | 0 | 0 | 0 |
| Indiana A | Northwestern A | 10 | 10 | 10 | 30 |
| Rutgers B | Vanderbilt A | 0 | 0 | 0 | 0 |
| Florida A | Stanford A | 10 | 10 | 0 | 20 |
| Purdue A | UC Berkeley B | 0 | 0 | 0 | 0 |
| Georgia Tech A | Cornell A | 10 | 10 | 0 | 20 |
| Yale B | WUSTL B | 10 | 0 | 0 | 10 |
| Chicago B | Brown A | 10 | 10 | 0 | 20 |
| Duke A | Maryland A | 0 | 0 | 0 | 0 |