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{\Huge 189-235A: Basic Algebra I} \\ \sk
{\Huge Midterm Exam} \\ \sk
{\Large Friday, October 22 }
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\def\Z{{\bf Z}}
\def\C{{\bf C}}
\def\R{{\bf R}}
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\noindent
1. Let $(u_n)_{n\ge 0}$ be the sequence of real numbers defined recursively
by the rule
$$ u_0 = 0, \quad u_{n+1} = 2 u_n + 1. $$
Show that $u_n = 2^n-1$ for all $n\ge 0$.
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\noindent
2. Compute the greatest common divisor of $121$ and $77$ and express the result
as a linear combination of $121$ and $77$.
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3. Solve the congruence equation
$6x\equiv 10 \pmod{14}.$
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4. Show that if $p\in \Z$ is a prime, then the ring $\Z_p$ of congruence
classes modulo $p$ is a field.
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5. Give an example of two finite rings $R_1$ and
$R_2$ which have the same cardinality but are not
isomorphic. (You should justify your assertion.)
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6. Show that the ring $\C$ of
complex numbers is {\em not} isomorphic to the Cartesian product $\R\times\R$
of the real numbers with itself.
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{\bf The next two problems are Bonus Questions}
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7.
Let $f$ be a polynomial in $\Z[x]$ of degree $d$
and let $p\in \Z$ be a prime number.
Show that the set
$$ S = \{ n\in \Z \mbox { such that } p \mbox{ divides } f(n) \}$$
is the union of at most $d$ congruence classes modulo $p$.
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8. Let $p = 2m+1$ be an odd prime.
Show that
$$ 1^1 \cdot 2^2 \cdot 3^3 \cdots (p-1)^{p-1}
\equiv (-1)^{[m/2]} m! \pmod{p}.$$
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