189-456A: Algebra 3

Assignment 5

Due: Wednesday, November 20.

1. Recall that a linear transformation $T:V \rightarrow V$ is said to be idempotent if $T^2=T$. If $T$ is such an idempotent and $V$ is finite-dimensional, show that $V$ is the direct sum of the kernel of $T$ and the image of $T$, and that $T$ is diagonalisable.

2. Let $V$ be a finite-dimensional vector space over a field $F$, and let $T$ be a linear endomorphism of $V$ of (exact) order $2$, satisfying $T^2=1$. Show that $T$ is always diagonalisable when $1+1\ne 0$ in $F$, and that it is never diagonalisable when $1+1=0$. Give an example of a matrix of order $2$ in ${\rm GL}_2(\mathbb Z/2\mathbb Z)$.

3. Let $T$ be a linear endomorphism of a finite-dimensional vector space $V$ over $F$. Show that $V$ has a cyclic vector for $T$ if and only if the degree of the minimal polynomial of $T$ is equal to the dimension of $V$.

4. Let $p$ be a rational prime and let $F$ be the field with $p$ elements. Show that $F[x]/(x^2+1)$ is isomorphic to $F[\varepsilon]/(\epsilon^2) $ when $p=2$, to $F\times F$ when $p$ is of the form $1+4k$, and to a field of cardinality $p^2$ when $p$ is of the form $3+4k$.

5. Let $T_1$ and $T_2$ be two linear transformations from $V$ to itself having the same minimal polynomial $p(x)$. Show that $T_1$ and $T_2$ are conjugate to each other when the degree of $p(x)$ is equal to the dimension of $V$. Give an example of two non-conjugate linear tranformations with the same minimal polynomial, to show that the assumption on the degree of $p(x)$ is necessary for the validity of this statement.

6. Let $p$ be a prime and let $F$ be the field with $p$ elements. Show that ${\rm GL}_2(F)$ contains a unique conjugacy class of elements of (exact) order $4$ when $p$ is of the form $3+4k$, and that it contains seven such conjugacy classes when $p$ is of the form $1+4k$. (Hint: the insights gained in questions $4$ and $5$ should be useful.)

7. A linear endomorphism $T$ of a vector space $V$ is said to be nilpotent if $T^n=0$ for some $n$. Show that the group $G={\rm Aut}_F(V)$ acts by conjugation on the set $X$ of nilpotent transformations on $V$, and that there are finitely many orbits for this action. Show that the number of orbits is equal to the number of conjugacy classes in the symmetric group $S_n$, where $n={\rm dim}_F(V)$. (Hint: think of the Jordan canonical form.)

8. Let $R$ be the ring $\mathbb Z[\sqrt{-5}]$ generated by $\sqrt{-5}$ over $\mathbb Z$. Show that the elements $2$, $3$, $(1+\sqrt{-5})$ and $(1-\sqrt{-5})$ are all irreducible in this ring, and conclude that $R$ is not a unique factorisation domain (namely, a ring in which every element can be uniquely expressed as a product of irreducible elements, up to units).

9. If $I$ and $J$ are two ideals in a ring $R$, their product $IJ$ is the ideal generated by elements of the form $ab$ with $a\in I$ and $b\in J$. Recall that $(a,b) = Ra + Rb$ is our notation for the ideal generated by $a$ and $b$ in $R$. Show that, in the ring $R=\mathbb Z[\sqrt{-5}]$, we have $$ (2,1+\sqrt{-5}) (2,1-\sqrt{-5}) = (2,1+\sqrt{-5})^2 = (2),\qquad (3,1+\sqrt{-5})(3,1-\sqrt{-5}) = (3),$$ $$ (2,1+\sqrt{-5})(3,1+\sqrt{-5}) = (1+\sqrt{-5}), \qquad (2,1-\sqrt{-5})(3,1-\sqrt{-5}) = (1-\sqrt{-5}).$$ Use this express the ideal $(6)$ as a product of prime ideals in $R$.

10. Let $F$ be a finite field of odd cardinality $q$. Show that the function $\varphi:F^\times\rightarrow F^\times$ given by $\varphi(x)=x^2$ is a group homomorphism. What is its kernel? What is the cardinality of its image? Use this to construct a field of cardinality $q^2$.