Exam
ALGEBRA PRELIM EXAM August 16, 2004
Fall 2004 · 10 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Set \[ J= \begin {pmatrix} 0&1&0&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&1&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0 \end {pmatrix} \in M_{10}(\mathbb {C}); \] here it is understood that unless otherwise noted, the entries of \(J\) are 0. Let \[ S=\{X\in M_{10}(\mathbb {C})\mid X^2=gJg^{-1}\text { for some }g\in \operatorname {GL}(10,\mathbb {C})\}. \] Observe that \(\operatorname {GL}(10,\mathbb {C})\) acts on \(S\) by conjugation. Compute the number of orbits for this action.
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Question 2
Let \(R\) be a commutative ring with identity. Given an \(R\)-module \(M\), recall the \(R\)-module \[ \bigwedge M=\bigoplus _{j\ge 0}\bigwedge ^j M, \] where \(\bigwedge ^j M\) is the \(j\)th exterior power of \(M\). Define a natural operation on morphisms so that the assignment \(M\mapsto \bigwedge M\) becomes a functor and determine whether this functor is exact.
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Question 3
Let \(V\) be a finite-dimensional vector space over a field \(k\), and let \(A\) and \(B\) be diagonalizable linear operators on \(V\) such that \(AB=BA\). Prove that \(A\) and \(B\) are simultaneously diagonalizable; i.e. that there is a basis for \(V\) consisting of eigenvectors of both \(A\) and \(B\).
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Question 4
Let \(R\) be a commutative ring with identity. Let \(G\) be a finite subgroup of \(R^\times \), the group of units of \(R\). Prove that if \(R\) is an integral domain, then \(G\) is cyclic.
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Question 5
Suppose \(R\) is a commutative Noetherian ring. Let \(R[[x]]\) denote the ring of formal power series with coefficients in \(R\). Prove that \(R[[x]]\) is Noetherian.
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Question 6
Prove that there are no simple groups of order 616.
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Question 7
Let \(F\) be a finite field, and suppose that the subfield of \(F\) generated by \(\{x^3\mid x\in F\}\) is different from \(F\). Show that the cardinality of \(F\) is 4.
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Question 8
Prove or disprove that \(f(x)=x^4+x^3+x^2+6x+1\) is irreducible over \(\mathbb {Q}\).
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Question 9
Prove that the Galois group of \(x^3+x-1\) over \(\mathbb {Q}\) is the symmetric group on three letters.
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Question 10
Let \(G\) be a finite group with identity \(e\). Suppose that for all \(a,b\in G\) distinct from \(e\), there exists an automorphism \(\tau \) of \(G\) such that \(\tau (a)=b\). Prove that \(G\) is abelian.