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\begin{document}
\subsection{Review Exercise}

Write the definition of each of the following terms or state the appropriate
theorem. Neatness counts! Use Scientific Workplace (available in the Math Lab).

\begin{enumerate}
\item relation

\item function

\item equivalence relation

\item equivalence class

\item the image of a set under a function

\item the range of a function

\item divides for integers (i.e. define $a|b$ )

\item The Division Algorithm for integers

\item gcd for integers

\item lcm for integers

\item The Euclidean Algorithm for integers

\item prime number

\item The Fundamental Theorem of Arithmetic

\item math induction

\item ring

\item commutative ring

\item ring with identity

\item integral domain

\item field

\item subring

\item unit

\item associates in a commutative ring with identity

\item zero divisor

\item ring homomorphism

\item isomorphism

\item polynomial

\item degree of a polynomial

\item monic polynomial

\item $R\left[  x\right]  $ where $R$ is a ring

\item divides in $F\left[  x\right]  $ where $F$ is a field (i.e. define
$f\left(  x\right)  |g\left(  x\right)  $ )

\item The Division Algorithm for polynomials in $F\left[  x\right]  $

\item gcd in $F\left[  x\right]  $

\item irreducible polynomial in $F\left[  x\right]  $

\item ideal

\item kernel of a ring homomorphism

\item quotient ring

\item coset

\item first isomorphism theorem for rings

\item group

\item order of a group

\item order of an element in a group

\item subgroup

\item cyclic group

\item the subgroup generated by a subset $S$ of a group $G.$

\item finitely generated group

\item Abelian group

\item group homomorphism

\item group isomorphism

\item normal subgroup

\item quotient group

\item kernel of a group homomorphism

\item first isomorphism theorem for groups

\item Lagrange's Theorem

\item simple group

\item permutation

\item symmetric group

\item alternating group
\end{enumerate}


\end{document}