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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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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