Exercises 4.11 Exercises
2.
Find \(a,b\in\R\) such that \(X^3+3X+4\equiv a+bX\in\R[X]/(X^2+1)\text{.}\)3.
Show that the set \(\{f\in\C[X,Y]:f(a,a)=0\;\text{for all}\;a\in\C\}\) is a principal ideal in \(\C[X,Y]\text{.}\) Furthermore show that this ideal is a prime ideal.
Hint.
Consider the polynomial \(X-Y\) and use the division algorithm.
4.
Show that \(X^2+1\) (resp. \(X^2+X+1\)) is irreducible in \(\Q[X]\) and \(\R[X]\text{.}\) Also show that \(\Q[X]/(X^2+1)\) and \(\R[X]/(X^2+1)\) (resp., \(\Q[X]/(X^2+X^2+1)\) and \(\R[X]/(X^2+X+1)\)) are fields. Further show that \(\R[X]/(X^2+1)\simeq\R[X]/(X^2+X+1)\text{.}\)5.
Let \(\mathbb{F}_2\) be the field with two elements. Show that \(\mathbb{F}_2[X]/(X^2+X+1)\) is a field.6.
Show that \(\R[X]/(X^2+1)\) and \(\R[X]/(X^2+X+1)\) (resp., \(\Q[X]/(X^2+1)\) and \(\Q[X]/(X^2+X+1)\)) are finite-dimensional vector spaces over \(\R\) (resp., \(\Q\)). Also show that \(\mathbb{F}_2[X]/(X^2+X+1)\) is a finite-dimensional vector space over \(\mathbb{F}_2\text{.}\)7.
Let \(A\) be a commutative ring and let \(a\in A\text{.}\) Show that \(A[X]/(X-a)\simeq A\text{.}\)8.
Let \(F\) be a field and let \(F[[t]]\) be the formal power series ring. Show that \(\mfm=\{\sum_{t\geq 0} a_it^i:a_0=0\}\subset F[[t]]\) is an ideal generated by \(t\text{.}\) Further show that \(\mfm\) is the only maximal ideal in \(F[[t]]\text{.}\)
Hint.
Refer to an elaborate discussion done in the class.
9.
Let \(F\) be a field and consider the following subset of \(M_n(F)\text{.}\)
\begin{equation*}
R=\left\{\begin{pmatrix}
a\amp0\amp0\amp\cdots\amp0\\a_{21}\amp a\amp0\amp\cdots\amp0\\a_{31}\amp a_{32}\amp a\amp\cdots\amp0\\\vdots\amp\vdots\amp\vdots\amp\ddots\amp\vdots\\a_{n1}\amp a_{n2}\amp a _{n3}\amp\cdots\amp a
\end{pmatrix}:a, a_{ij}\in F\right\}
\end{equation*}
Show that \(R\) is a ring and that \(I=\{A\in R:\det(A)=0\}\) is the only maximal ideal of \(R\text{.}\)
Hint.
Show that every element not in \(I\) is invertible.
10.
Use the Chinese remainder theorem to find a solution to the following congruences
\begin{align*}
x\amp\equiv 2\mod 5\\
x\amp\equiv 3\mod 7\\
x\amp\equiv 10\mod 11
\end{align*}
