189-456A: Algebra 3

Practice Assignment 6

Due: Wednesday, December 4.

1. Let $T$ be a linear transformation over a field $F$ having $(x-\lambda)^2$ as minimal polynomial, for some $\lambda\in F$, and let $g(x)$ be a polynomial in $F[x]$. Show that $$g(T) = g(\lambda) I + g'(\lambda) (T-\lambda I),$$ where $I$ is the identity transformation. Can you generalise this formula to the case where the minimal polynomial is $(x-\lambda)^k$?

2. Show that an element of $\mathbb Z[i]$ is irreducible if and only if it is either of the form $i^k p$ where $p$ is a rational prime of the form $4k+3$, or of the form $a+bi$ where $a^2 + b^2$ is either $2$ or a rational prime of the form $4k+1$.

3. Factor the element $-31+51i$ into prime elements of $\mathbb Z[i]$.

4. Let $R=\mathbb Z[\alpha]$ be the ring generated over $\mathbb Z$ by a complex number $\alpha$ satisfying the polynomial $x^2+x+6 = 0$. Show that the ideal $I=(2,\alpha)$ generated by $2$ and $\alpha$ is not principal, and that the same is true for $I^2$, but that $I^3$ is a principal ideal. What is a generator of this ideal?

5. The regular icosahedron is a regular solid in three-dimensional space whose faces are isosceles triangles. The group of rotations which preserve this figure is isomorphic to the alternating group $A_5$ on five elements, and it acts transitively on the edges, vertices, and faces of the icosahedron. Each vertex is contained in five faces, and every face is preserved by a rotation of order $3$. From this information, compute the number of faces, edges and vertices in the regular icosahedron. (A competent latinist might guess at the answer, but please indicate a mathematical reasoning!)

6. Let $T:V\rightarrow V$ be a linear transformation on a finite-dimensional vector space $V$ over a field $F$. Show that the set of linear transformations that commute with $T$, i.e., satisfy $ST = TS$, is a subring of the ring ${\rm End}_F(V)$. Give a necessary and sufficient condition on $T$ for this ring to be commutative.

7. Let $G=GL_3(F_2)$ be the group of order $168$ consisting of the invertible $3\times 3$ matrices with coefficients in the field with $2$ elements. Describe all the conjugacy classes in $G$ and their sizes, and write down the class equation for $G$. (Partial elements of solution for this have been worked out in previous assignments, and a full treatment is given in Garret's book, but make sure you spend some serious time engaging with the problem in front of a blank piece of paper before taking a peak!)

8. Describe a Sylow $3$-subgroup of $GL_3(F_p)$ where $F_p$ is the field with $p$ elements and $p$ is a prime of the form $1+3k$ with $k$ not divisible by $3$.

9. Let $R$ be a principal ideal domain. Show that there is no infinite strictly increasing sequence of ideals $I_1\subset I_2 \subset \cdots$ ordered by inclusion.

10. An $R$-submodule $N$ of an $R$-module $M$ is said to be a direct summand in $M$ if there is a submodule $N'$ of $M$ with $M = N \oplus N'$. Let $R$ be a PID. Show that an $R$-submodule $N$ of a finitely generated free $R$-module $M$ is a direct summand in $M$ if and only if the quotient $M/N$ is free over $R$.