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  {\bf Math 113 Homework 12, due 4/23/2019.}
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  {\bf Note that the due date is the day of the midterm -- it is strongly recommended that you do these exercises by Thursday, 4/18.}
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\noindent {\bf 1.} Let $f(x) = x^2 + x + 1\in \Z_7[x].$ Compute the residue of $g(x) = x^4 + 3x^3 + 3x^2 + x + 4$ modulo $f$. 

\noindent {\bf 2.} Let $f(x) = x^2 +1\in \R[x].$

(a) Compute the residue of $s(x) = a + bx + cx^2$ modulo $f.$ (It should be a polynomial of degree less than $f$.)

(b) Compute the product $([a + bx])([c + dx])\in \R[x]/(f),$ by finding the residue of the product modulo $x^2 + 1$.

(c) Show that the function  $\R[x]/(f)$ is isomorphic to the complex numbers via the homomorhpism $\phi:\R[x]\to\C$ given by
$\phi([a + bx]): = a + bi.$

\noindent {\bf 3.} Let $g(x) = x^3 -2x -4\in \Q[x].$ Check that $[x-2]$ is a zero divisor in $\R[x]/(g).$ In other words, there is a nonzero remainder $f(x)$ (of degree smaller than $g$) such that $f(x)*(x-2)$ is equal to $0$ modulo $g(x).$ (Hint: use the division algorithm to divide $g(x)$ by $x-2$.)




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