Question
According to Maschke’s theorem, every group representation of a finite group over a field with characteristic zero is the direct sum of representations with this property. For 10 points each:
[10h] Name this adjective. Gauss’s lemma states that any polynomial with a property over the rational numbers will have this property over the integers.
ANSWER: irreducible [or word forms; accept not reducible or equivalents; reject “reducible”]
[10m] Gauss’s lemma applies to a unique factorization domain, which is an example of one of these structures. These algebraic structures generalize fields and are closed under their addition and multiplication operations.
ANSWER: rings
[10e] A different result also named Gauss’s lemma provides conditions for an integer to be a residue known by this adjective. This adjective denotes polynomials of degree 2, like “a x-squared plus b x plus c.”
ANSWER: quadratic [accept quadratic residues or quadratic polynomial or quadratic equation or quadratic form]
<Other Science>
Summary
| 2024 ACF Winter at UC Berkeley | 2024-11-16 | Y | 3 | 16.67 | 67% | 67% | 33% |
| 2024 ACF Winter at Clemson | 2024-11-16 | Y | 8 | 12.50 | 88% | 38% | 0% |
| 2024 ACF Winter at Northwestern | 2024-11-16 | Y | 9 | 16.67 | 100% | 44% | 22% |
| 2024 ACF Winter at Ohio State | 2024-11-16 | Y | 7 | 14.29 | 100% | 43% | 0% |
| 2024 ACF Winter at Online | 2024-11-16 | Y | 8 | 13.75 | 100% | 38% | 0% |
| 2024 ACF Winter at UBC | 2024-11-16 | Y | 3 | 16.67 | 100% | 67% | 0% |
| 2024 ACF Winter at Oxford | 2024-11-16 | Y | 11 | 15.45 | 91% | 46% | 18% |
Data
| UIUC B | Indiana B | 0 | 0 | 10 | 10 |
| Miami | Purdue B | 0 | 0 | 10 | 10 |
| Northwestern A | UIUC C | 0 | 10 | 10 | 20 |
| Indiana A | Notre Dame | 0 | 10 | 10 | 20 |
| WashU B | Purdue A | 10 | 10 | 10 | 30 |
| Purdue C | UChicago A | 0 | 0 | 10 | 10 |
| WashU D | Purdue D | 0 | 0 | 10 | 10 |
| UChicago B | SIUE A | 10 | 10 | 10 | 30 |
| UIUC D | WashU C | 0 | 0 | 10 | 10 |