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\hspace*{\fill} December 7, 1999
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{\bf MA COMPREHENSIVE EXAM IN ALGEBRA}
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Do {\bf two} problems from each of the three parts. Please give complete proofs. If you do three problems in one of the parts, please indicate which two problems you want graded.
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{\bf Part A: Groups}
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1. Let $a, b$ and $c$ be elements of a group $G$. For each of the following statements, give either a proof or a concrete counterexample.
(a) If $a$ has order 5 and $a^{3}b=ba^{3}$, then $ab=ba$.
(b) If $abc=1$, then $cab=1$.
(c) If $abc=1$, then $bac=1$.
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2. If $G$ is a group and $H$ is a subgroup of $G$ of index $n$, show that $G$ contains a {\it normal} subgroup $K$ whose index divides $n!$.
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3. Show that every group of order 1225 is abelian.
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{\bf Part B: Rings}
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4. (a) Show that there is no polynomial $p(x) \in {\bf Z}[x]$ with $5p(x)=1$.
(b) Find a ring $R$ and an explicit ring homomorphism $\phi : {\bf Z}[x] \rightarrow R$ such that $\phi$ is onto and such that there exists an element $a \in R$ with $5a=1$.
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5. Let $R$ be a U.F.D. (unique factorization domain) and assume that each of $f(x), g(x) \in R[x]$ has content 1. Prove that $f(x)g(x)$ has content 1.
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6. (a) Recall that an ideal $P$ in a commutative ring is {\it prime} if, whenever $x, y \in R$ such that $xy \in P$, either $x \in P$ or $y \in P$.
Show that in ${\bf Q}[x]$, every prime ideal is a maximal ideal.
(b) Exhibit a prime ideal in ${\bf Z}[x]$ which is {\it not} maximal.\\
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{\bf Part C: Linear Algebra}
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7. Let $U$ and $W$ be subspaces of the finite dimensional vector space $V$. Prove that $dim U + dim W = dim (U+W) + dim (U \cap W)$.
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8. If $V$ is a finite dimensional inner product space and $U$ is a subspace of $V$, let $U^{\perp}$ be the orthogonal complement of $U$. Prove that
(a) $V = U \bigoplus U^{\perp}$
(b) $U^{\perp \perp} = U$.
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9. Let $V$ be the space of all polynomials in $x$ of degree at most 3 over ${\bf R}$, with the standard basis $B = \{1, x, x^2, x^3 \}$. Let $T$ be the linear operator on $V$ defined by $$T(f) = f + \frac{df}{dx}.$$
Find
(a) the matrix of $T$ with respect to the basis $B$
(b) the characteristic and minimal polynomials of $T$
(c) the rank and nullity of $T$
(d) bases for the range and nullspace of $T$
(e) all eigenvalues of $T$.
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