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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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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