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

Data

Berkeley ABerkeley B10101030
Stanford AStanford L0000
Stanford MBerkeley C0101020
Georgia AAlabama A001010
Georgia Tech DAuburn A0101020
Tennesse BClemson A0000
Emory AAuburn C001010
Georgia Tech EGeorgia Tech B001010
Georgia Tech FAuburn B0101020
Emory BSouth Carolina A001010
Tusculum AGeorgia Tech C0101020
UIUC BIndiana B001010
MiamiPurdue B001010
Northwestern AUIUC C0101020
Indiana ANotre Dame0101020
WashU BPurdue A10101030
Purdue CUChicago A001010
WashU DPurdue D001010
UChicago BSIUE A10101030
UIUC DWashU C001010
CWRU A (UG)CWRU C (UG)0101020
Kenyon AOSU C001010
Michigan AMichigan State A0101020
Michigan BCWRU D (DII)001010
Michigan C (UG)Michigan State C (UG)001010
CWRU B (UG)Michigan D (DII)001010
Pitt AOhio State A (UG)0101020
Texas BArkansas0101020
Vassar ACentral Oklahoma001010
Texas AColorado College0101020
IowaTexas D001010
McGill ENYU B0101020
Mississippi StateMissouri001010
Ole MissOregon State001010
Texas CNYU A001010
AlbertaUW B0101020
UBC BSFU0101020
UW AUBC A001010
Cambridge AImperial A10101030
Bristol ACambridge B001010
Cambridge CLSE A001010
Durham ACam D001010
Bristol BEdinburgh001010
Imperial BOxford B0101020
Oxford AManchester10101030
Southampton ABirmingham0101020
VanderbiltLSE B0000
Warwick ASouthampton B0101020
Durham BWarwick B001010
A Brandeis SupremeClark A001010
Amherst ABoston University A0101020
Bowdoin BDiamond Brandeis001010
Brandeises BrewYale C001010
Brown AMIT A10101030
Yale ACarabrandeis0101020
Harvard ANortheastern A0101020
Tufts ATufts B001010
Williams ABowdoin A0101020
Yale BBoston University B001010