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{\Huge 189-235A: Basic Algebra I} \\ \sk
{\Huge Practice Final Exam} \\ \sk
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\def\Z{{\bf Z}}
\def\C{{\bf C}}
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\def\Q{{\bf Q}}
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\def\GL{{\bf GL}}
\def\lra{{\longrightarrow}}
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{\em This exam has ten
questions,
worth 10 points each.
The bonus question is worth 20 points.
The final grade will be out of $100$, even though the maximum
possible grade is $120$. }
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\noindent
1. Show that $64$ divides $9^n-8n-1$ for every $n\ge 0$, by induction on $n$.
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2. Solve each of the following congruence equations.
a) $6x=3 \pmod{30},$
b) $x^2 + 4x+5 = 0 \pmod{13}. $
c) $x^7=1 \pmod{101}$.
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3. The {\em exponent } of a finite group is the smallest positive integer
$n$ such that $a^n=1$ for all $a\in G$.
Let $G$ be an abelian group.
a) Show that $G$ is cyclic if and only
if its exponent is equal to its cardinality.
b) Use part a) to show that a finite subgroup of the multiplicative group
of a field is necessarily cyclic.
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\noindent
4. Prove that the rings $\R[x]/(x^2+1)$ and $\R[x]/(x^2-1)$
are {\em not} isomorphic.
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5. Give a non-commutative group $G$ of order $125$ in which
$a^5=1$ for all $a\in G$.
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6. Are the groups $D_4\times \Z_3$ and $S_3\times \Z_4$
isomorphic? Prove or disprove.
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7. Give the definitions of: {\em maximal
ideal} and {\em prime ideal} in a commutative ring
$R$.
Show that if $I$ is a prime ideal then $R/I$ is an integral domain.
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8. State the isomorphism theorem for rings.
Show that the ring $R_1= \Z[x]/(3x-2)$ is isomorphic to the subring $R_2$
of $\Q$ consisting of all rational numbers whose denominator is a power of
$3$. (Hint: construct a surjective homomorphism from $\Z[x]$ to
$R_2$ whose kernel is the ideal $(3x-2)\Z[x]$.)
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9. For each of the following rings $R$, state whether or not there exists
a homomorphism from $R$ to $\Z_3$, and, if so, how many there are.
9a) $R=\Z_7$;
9b) $R=\Z_3[x]$;
9c) $R = \Z_3[x]/(x^2+2)$;
9d) $R = \Z_3[x]/(x^2-2)$.
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10. Show that the symmetric group $S_5$ contains elements of order $6$, and that
they are all conjugate. How many such elements are there?
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{\bf Extra Credit Problem}
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11. Let $G=\GL_3(\Z_2)$ be the group of invertible $3\times 3$ matrices with
entries in $\Z_2$. What is the cardinality of this group?
Write down representatives for each of the distinct conjugacy classes,
and give their orders, ie., compute the class equation for $G$.
Use this to show that $G$ has no non-trivial normal subgroups.
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