A Master's in motherhood?

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Masters Examination in Mathematics

Solution: The ring F [x] of polynomials with coefficients in a field F is a P.I.D. Each prime ideal is generated by a monic, irreducible polynomial. Assume there are only a finite number of prime ideals generated by the polynomials f1, . . . , fn and let f(x) = 1+f1(x) · · · fn(x). No fi divides f , hence f is also irreducible. This contradicts the assumption that all the prime ideals were gene...

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Masters Examination in Mathematics

Full points can be obtained for correct answers to 8 questions. Each numbered question (which may have several parts) is worth 20 points. All answers will be graded, but the score for the examination will be the sum of the scores of your best 8 solutions. Algebra A1. (a) State the Class Equation.

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Solution. Since (ab)(abc) = (bc) and (abc)(ab) = (ac), it is easy to see that the center of S3 is the trivial subgroup. Therefore, the group of inner automorphisms of S3 is isomorphic to S3 and has size 6. On the other hand, S3 has 3 transpositions. These must be permuted by an automorphism of S3, and a nontrivial automorphism induces a nontrivial permutation. This gives an injective homomorphi...

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Solution. Let np denote the number of p-Sylow subgroups. Since n11 is 1 mod 11 and divides 12, we know n11 is either 1 or 12. If it is 1, then it is normal and we are done. So instead, assume n11 is 12. Then there must be 120 elements of order 11 in G, leaving only 12 more elements available. Since n3 is 1 mod 3 and divides 44, we know that n3 is 1, 4 or 22. If it is 1, then again it is normal ...

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ژورنال

عنوان ژورنال: Work

سال: 2015

ISSN: 1051-9815

DOI: 10.3233/wor-141952