FIN - polynomial f ( x ) which generates the principal...

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Math 113 (Bernd Sturmfels), Final Exam Thursday, August 17, 8:10 a.m.–10:00 a.m. Please start by writing your name and your student ID on the cover of your blue book. This exam is closed book . Do not use any notes, calculators, cell phones etc. You must show all your work to get credit. There are ten problems, each worth 10 points, for a total of 100 points. (1) Let u be the class of 11 in Z / 23 Z . Determine u - 1 and u + u - 1 . (2) What is the smallest order of a non-abelian group ? (3) How many units does the ring Z / 60 Z have ? (4) Determine the 11 th cyclotomic polynomial Φ 11 ( x ). (5) For which values of a in F 5 is the ring F 5 [ x ] / h x 3 + 2 x 2 + a i a field ? (6) Consider the ideal I = h x 2 + y 2 - 1 ,xy + 2 i in Q [ x,y ]. Find a
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Unformatted text preview: polynomial f ( x ) which generates the principal ideal I Q [ x ] in Q [ x ]. (7) Prove or disprove: If H 1 and H 2 are subgroups of a nite group G then their product H 1 H 2 is a subgroup of G . (8) Prove or disprove: If I is an ideal in a principal ideal domain R then every ideal in the quotient ring R/I is principal. (9) Prove or disprove: There exists a term ordering such that the subset { x 3 + y 2 ,x 2 + y } of Q [ x,y ] is a Grobner basis. (10) Prove or disprove: If R is the quotient ring Q [ x,y, z ] / h x z-y 2 i then every irreducible element in R is prime....
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