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(MATH551)7a83d5 - final511.pdf
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FINAL EXAM OF MATH511
DUE BY 11/12
1.(20pt) Prove or disprove the following statements.
(1)
Any prime ideal of a commutative ring contains a minimal prime ideal.

(2)
Any torsion free module over a PID is a free module.

(3)
Let V be a complex vector space. Then V .C V = S2(V ) 2(V ).

(4)
Q is a .at Z-module.
2.(5pt) Find all positive integers n such that x4 + n is irreducible in Q[x].
3.(5pt) Let G be a group of order 825 = 33 25. Find all the composition factors



of G. 4.(5pt) Let V be the module over C[x] which is the 2-dimensional complex vector
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space with the action of x given by (with respect to some basis of V ).
02 Describe the ring HomC[x](V, V ).
5.(5pt) A square matrix M is called nilpotent if Mn = 0 for some n N. Let p be a prime and Fp be the .nite .eld with p elements. Prove that the number of nilpotent matrices in M22(Fp) equals p2 . (Hint: if M is a nilpotent matrix in M22(Fp), then M2 = 0.)
Bonus problem: Count the number of nilpotent matrices in M33(Fp).
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