irreducible - 2024 ACF Winter

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]

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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%
2024 ACF Winter at Brandeis	2024-11-16	Y	10	16.00	100%	50%	10%

Data


Berkeley A	Berkeley B	10	10	10	30
Stanford A	Stanford L	0	0	0	0
Stanford M	Berkeley C	0	10	10	20
Georgia A	Alabama A	0	0	10	10
Georgia Tech D	Auburn A	0	10	10	20
Tennesse B	Clemson A	0	0	0	0
Emory A	Auburn C	0	0	10	10
Georgia Tech E	Georgia Tech B	0	0	10	10
Georgia Tech F	Auburn B	0	10	10	20
Emory B	South Carolina A	0	0	10	10
Tusculum A	Georgia Tech C	0	10	10	20
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
CWRU A (UG)	CWRU C (UG)	0	10	10	20
Kenyon A	OSU C	0	0	10	10
Michigan A	Michigan State A	0	10	10	20
Michigan B	CWRU D (DII)	0	0	10	10
Michigan C (UG)	Michigan State C (UG)	0	0	10	10
CWRU B (UG)	Michigan D (DII)	0	0	10	10
Pitt A	Ohio State A (UG)	0	10	10	20
Texas B	Arkansas	0	10	10	20
Vassar A	Central Oklahoma	0	0	10	10
Texas A	Colorado College	0	10	10	20
Iowa	Texas D	0	0	10	10
McGill E	NYU B	0	10	10	20
Mississippi State	Missouri	0	0	10	10
Ole Miss	Oregon State	0	0	10	10
Texas C	NYU A	0	0	10	10
Alberta	UW B	0	10	10	20
UBC B	SFU	0	10	10	20
UW A	UBC A	0	0	10	10
Cambridge A	Imperial A	10	10	10	30
Bristol A	Cambridge B	0	0	10	10
Cambridge C	LSE A	0	0	10	10
Durham A	Cam D	0	0	10	10
Bristol B	Edinburgh	0	0	10	10
Imperial B	Oxford B	0	10	10	20
Oxford A	Manchester	10	10	10	30
Southampton A	Birmingham	0	10	10	20
Vanderbilt	LSE B	0	0	0	0
Warwick A	Southampton B	0	10	10	20
Durham B	Warwick B	0	0	10	10
A Brandeis Supreme	Clark A	0	0	10	10
Amherst A	Boston University A	0	10	10	20
Bowdoin B	Diamond Brandeis	0	0	10	10
Brandeises Brew	Yale C	0	0	10	10
Brown A	MIT A	10	10	10	30
Yale A	Carabrandeis	0	10	10	20
Harvard A	Northeastern A	0	10	10	20
Tufts A	Tufts B	0	0	10	10
Williams A	Bowdoin A	0	10	10	20
Yale B	Boston University B	0	0	10	10