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{\Huge 189-235A: Basic Algebra I} \\ \vskip 0.1in
{\Huge Assignment 5} \\ \vskip 0.1in
{\Large Due: Thursday, October 20 }
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1. Perform the division algorithm
for dividing $f(x) = 3x^4-2x^3+6x^2-x+2$ by $g(x) = x^2+x+1$ in
$\Q[x]$.
(I.e.,
find polynomials $q(x)$ and
$r(x)$ with ${\rm deg}(r)<{\rm deg}(g)$ satisfying
$f = gq + r.$
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2. Same question as 1, with
$f(x) = x^5-x+1$ and $g(x) = x^2+x+1$ in ${\bf Z}_2[x]$.
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3. Let $f:\Z[x]\rightarrow \Z$ be the function which
to any polynomial $p(x)=a_0+a_1 x + \cdots + a_d x^d$
associates its constant term $a_0$:
$f(p) = a_0$.
Show that $f$ is a homomorphism of rings.
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4. Find the gcd of
$x^4-x^3-x^2+1$ and $x^3-1$ in ${\bf Q}[x]$ using the Euclidean
algorithm.
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5. Find the gcd of
$x^4+3x^3- 2 x + 4$ and $x^2+1$ in ${\bf Z}_5[x]$ using the
Euclidean algorithm.
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6. Let ${\bf C}$ denote the field of complex numbers, and let
$\overline{z}=a-bi$ denote as usual
the complex conjugate of the complex number $z=a+bi$.
Let $H$ be the subset of $M_2(\C)$
consisting of matrices of the form
$$ \left(\begin{array}{cc}
z_1 & z_2 \\ -\overline{z_1} & \overline{z_2} \end{array} \right),$$
where $z_1$ and $z_2$ are complex numbers.
Show that $H$ is a subring of $M_2(\C)$.
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7. Show that the ring $H$ of exercise 6 is a non-commutative ring
in which every non-zero element has a multiplicative inverse.
(In other words, $H$ is a non-commutative field.)
The ring $H$ is called the field of {\em quaternions}.
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\noindent
{\bf The next few questions are optional}.
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8. Let $H'$ be the set of all expressions of the form
$a+bi+cj + dk$, where $a,b,c$ and $d$ are real numbers,
and $i$, $j$, $k$ are formal variables.
Define a multiplication on $H'$ by combining the usual rules for
addition and multiplication of real numbers with the rules
$$ i^2=j^2=k^2=-1, \quad ij = -ji =k, \quad ki=-ik=j, \quad jk=-kj=i.$$
Show that $H'$ is isomorphic to the ring $H$ of exercise 7.
(So from now on, we will write $H$ instead of $H'$.)
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9.
Let $H$ be the ring of real quaternions introduced in the previous
exercise.
A quaternion is said to be {\em integral} if it is of the form
$a 1 + b i + cj + dk$, where $a,b,c,d$ are {\em integers}.
Let $R$ be the set of integral quaternions.
Show that $R$ is a subring of $H$.
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\noindent
10. Define a ``complex conjugation" on $H$ by the rule
$$ \overline{a 1 + b i + c j + d k} := a 1 - b i - c j - d k.$$
Show that if $\alpha$ and $\beta$ are two quaternions, then
$$\overline{\alpha\beta} = \overline{\beta}\cdot \overline{\alpha}.$$
(Note the change in the order of multiplication!)
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\noindent
11.
If $\alpha$ belongs to $H$, define $||\alpha||:= \alpha\overline{\alpha}$.
Show that if $\alpha = a 1 + b i + c j + d k$, then
$$||\alpha|| = a^2+b^2+c^2+d^2.$$
Note in particular that if $\alpha$ belongs to the ring $R$ of integral
quaternions,
then $||\alpha||\in {\bf Z}$.
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12.
Show that $||\alpha\beta||=||\alpha||\cdot||\beta||.$
(Remember in your proof that multiplication in $H$ is not commutative!)
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13. Using 12, show that, if $m$ and $n$ are
integers which can be expressed as a sum of $4$ integer squares, then their
product $mn$ can also be expressed as a sum of four integer squares.
Use your proof to express $161=7\cdot 23$ as a sum of four squares.
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This last
exercise illustrates the usefulness of the ring $R$ for number theory, and
in particular for the study of representations of integers as sums of
four squares. A deeper study of the ring theoretic structure of
$R$ leads to the following beautiful theorem of Lagrange:
{\em Every positive integer can be expressed as a sum of four squares}.
You should try to test this theorem empirically to get a feeling for what
it says. Try also to find a proof!
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