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{\bf Math 113 Homework 11, due 4/16/2019}
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\noindent {\bf 1.} Find all $x$ with $x^2 - 4 = 0$ in $\Z_5\times \Z_7.$ Now find all $x$ with $x^2-4 = 0$ in $\Z_{35}$ (by the Chinese remainder theorem, the two sets of solutions have the same cardinality and get sent to one another under the isomorphism that takes $a\in \Z_{35}$ to $(a\mod 5, a\mod 7)$). 

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\noindent {\bf 2.} Recall that the Gaussian integers $\G$ are complex numbers $a+bi$ with integer $a, b$.

(a) Check that $\G\subseteq \C$ is a subring.

(b) Let $n\in \N$ be a positive integer. For $z, z'\in \G$ we say that $z\equiv_n z'$ if $n\mid z'-z$. Recall (from problem set 4) that every class in $\G/\equiv_n$ can be uniquely represented by $a + bi$ for $a, b\in \{0,\dots, n-1\}$ remainders modulo $n$. You have checked in earlier homeworks that addition and multiplication are well defined on $\G/\equiv_n$, hence it is also a ring. Check that $\G/\equiv_3$ is a field (by finding an inverse for every element). This is a field with $9$ elements --- in fact, any other field with $9$ elements is isomorphic to this one!

(c) Check that $\G/\equiv_5$ is not a field (hint: find a zero divisor).

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\noindent
{\bf 3.} 20.4, 20.5, 
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\noindent {\bf 4.} 20.8, 20.10 (the function $\phi(n)$ computes the number of residues mod $n$ which are relatively prime with $n$, so for example $\phi(p) = p-1$ for $p$ prime). 

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\noindent {\bf 5.} 20.23 (notation: ``unity'' means multiplicative identity element, ``unit'' means invertible element).

\noindent {\bf 6.} Extra credit: 20.27, 20.28



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