The questions that I assigned last year
as exercises might be interesting
solutions for exercises.
(2020/1/14)
For each of the following, determine if the construction is a functor of category or not.
1. $F\colon \mathbb{G}roup \to \mathbb{Ab G}roup$; $F\colon G\mapsto Z(G)$ the center of $G$.
2. For a fixed integer $n\ge0$, $F\colon \mathbb{Z}-\mathbb{M}od \to \mathbb{Z}-\mathbb{M}od$ by $F\colon A\mapsto A/nA$.
3. For a fixed set $X$, $F\colon \mathbb{G}roup \to \mathbb{G}roup$ by $F\colon G\mapsto G^X$ where $G^X$ is the group of
functions $f\colon X\to G$ with pointwise multiplications.
(2020/1/14)
Show that the functor $F: {\mathbb k}-{\mathbb V}ect \to {\mathbb k}-{\mathbb V}ect$
which sends $V$ to the dual vector space $V^\ast := Hom(V, {\mathbb k})$ is a
contravariant functor.
(2020/1/16)
(a) Consider examples Example 1.2.3, 1.2.4, 1.2.5, 1.2.6, 1.2.7, 1.2.8, 1.2.9,
1.2.12, 1.2.13, 1.2.14 in Basic Category Theory. Which of these
are faithful and which are full?
(b) Write down one example of a functor that is both full and faithful,
one that is full but not faithful, one that is faithful but not full,
and one that is neither. (BCT 1.2.28 p. 27)
Give an example of an adjoint functor (other than the adjoint to the forgetful
functor from the definition of free object) and a functor which doesn't have
and adjoint.
Give an example of a non-concrete category.
(Note: Reference an explanation on a blog about
a paper from 2004 that the category
of homotopy is such an example. This shows that using accepted definitions of
forgetful functor, it is unlikely that we will come up with a simple example.)
Give an example (which is not the previous exercise of a non-concrete category)
of a category which doesn't have free objects.
I gave the defintion of direct product in a category. Reverse the arrows
in the defintion in an appropriate manner and give the definition of
coproduct of a category. Prove that the coproduct in a category (if it
exists) is unique up to isomorphism.
Prove the isomorphism theorems for R-modules. How much of the proofs can
be done at the level of categories? In particular, what part of the statements
of the isomorphism theorems rely on definitions that exist on the level of
categories and not just for the category of R-Mod.
Are there any parts of the isomorphism theorems that can be proven in
for any category where these constructions exist?
(2020/1/23)
We showed exercise #16 and 17 of DF section 10.3 for two ideals
$A_1$, $A_2$. Finish the proof for sequences of ideals $A_1, A_2, \ldots, A_k$.
Let $n$ be an integer and
$n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k}$ be its prime
factorization. Show that
$${\mathbb Z}/n {\mathbb Z} \simeq {\mathbb Z}/p_1^{\alpha_1} \times
{\mathbb Z}/p_2^{\alpha_2} \times \cdots
{\mathbb Z}/p_k^{\alpha_k}$$
(2020/2/4)
Let $A_1, A_2, \ldots, A_n$ be $R$-modules and $B_i$ be a submodule of
$A_i$ for each $i = 1, \ldots, n$. Prove that
$$(A_1 \times \cdots \times A_n)/(B_1 \times \cdots \times B_n)
\simeq (A_1 / B_1) \times \cdots \times (A_n/B_n)$$
[DF, Ch 10.2, #11]
Find and example of a P.I.D. $R$ and elements $a, a', b, b' \in R$ such
that
$$R/(a) \oplus R/(a') \simeq R/(b) \oplus R/(b')$$
such that $(a)$ is not either $(b)$ or $(b')$.
[DF, 12.1, p.468 just before the exercises]
(2020/2/6)
Find a basis for $V = {\mathcal L}\{v_1, v_2, v_3\}$
such that the matrix for the linear transformation $T$ is in rational
canonical form where $T(v_1) = v_1 + 3 v_2 + v_3$, $T(v_2) = 2 v_2 + v_3$,
$T(V_3) = 4 v_1 + v_2$.
Let $A$ be the matrix
$$\left[\begin{matrix}0&-1&-2&-3\\1&0&0&0\\0&0&1&0\\1&0&0&1\\\end{matrix}\right]$$
(a) What is the rational canonical form of $A$ over ${\mathbb Q}$?
[Bonus: give the matrix $M$ such that $M^{-1} A M$ is the
rational canonical form].
(b) What is the Jordan form of $A$ over ${\mathbb C}$?
(2020/2/11)
Given the matrix
$$A = \left[\begin{matrix} -3& -8& -16\\2& 6 &13\\-1 &-3 &-6\end{matrix}\right]$$
find a matrix $P$ such that $PAP^{-1}$ is in rational canonical form
and a matrix $Q$ such that $QAQ^{-1}$ is in Jordan canonical form.
Hint:
$$V^{-1} = \left[ \begin{matrix}
1 & -x/2 + 3 & 13/2\\
0 & 1 &x^2 + 3 x + 5\\
0 & 0 & 1\end{matrix}\right]$$
where
$$U(xI - A)V = \left[\begin{matrix}
1& 0 & 0\\
0&1&0\\
0&0&x^3 + 3x^2 + 3x + 1\\
\end{matrix}\right]$$
(2020/2/13)
Let $P_1$ and $P_2$ be $R$ modules. Prove that $P_1 \bigoplus P_2$ is a projective
$R$ module if and only if both $P_1$ and $P_2$ are projective. [DF, p. 403, 10.5 #3]
(note, last year we were doing this problem and got stuck in one direction
and we got help from R. Sulzgruber)
Let $Q_1$ and $Q_2$ be $R$ modules. Prove that $Q_1 \bigoplus Q_2$ is an injective
$R$ module if and only if both $Q_1$ and $Q_2$ are injective. [DF, p. 403, 10.5 #4]
(2020/3/04)
Recall in the proof of the statement "P is projective $\Leftarrow$
for all surjections $\phi: M \rightarrow P$, there is a map $\psi : P \rightarrow M$
such that $\phi \circ \psi = id_P$" we showed that $\psi$ is injective
and $\psi(P) \cap ker~\phi = \{0\}$. Fill in the detail why
$y = (y - \psi(\phi(y))) + \psi(\phi(y))$ shows that $M = ker~\phi \oplus \psi(P)$.
Prove that if there is a free module $X$ such that $X = P' \oplus B$ for a
module $B$ and $P' \cong P$, then $P$ is projective.
Prove that if $P_1 \oplus P_2$ is projective, then $P_1$ is projective (yea, I know this
is a repeat from the one above, but we did one direction already).
Suppose that
is a commutative diagram of groups and the rows are exact. Prove that
(a) if $\phi$ and $\alpha$ are surjective and $\beta$ is injective,
then $\gamma$ is injective.
(b) if $\psi'$, $\alpha$, and $\gamma$ are injective, then $\beta$ is injective
(c) if $\phi$, $\alpha$, and $\gamma$ are surjective, then $\beta$ is surjective
(d) if $\beta$ is injective, $\alpha$ and $\gamma$ are surjective, then $\gamma$ is injective,
(e) if $\beta$ is surjective, $\gamma$ and $\psi'$ are injective, then $\alpha$ is surjective.
[DF, p.403, sec 10.5 # 1] (Note: the fact that this is a commutative diagram of groups is is important).
(2020/3/17-19)
Let $K$ be an extension field of $F$ and let $\alpha$ be an element of $K$.
Since $F[x]$ is a P.I.D., then the ideal
$I_\alpha = \{ p(x) \in F[x] : p(\alpha) = 0\}$ is equal to $\left< f(x) \right>$
for some $f(x) \in F[x]$. Prove that $f(x)$ is irreducible.
Give an example of a field $K$ which is an extension of ${\mathbb Q}$
which is algebraic but such that $[K : {\mathbb Q}]$ is not finite.
Let $p(x) = x^3 + 2x^2 - 10 \in {\mathbb Q}[x]$ and let $I = \left< p(x) \right>$.
(a) Let $E = {\mathbb Q}[x]/I$. Show that $E$ is a field.
(b) Compute $[E:{\mathbb Q}]$.
Find an example of a field $F$ and elements $\alpha,\beta$
such that $[F(\alpha):F] = n$ and $[F(\beta):F] = m$ for some
integers $m$ and $n$ and a field $K$ which contains
$F(\alpha)$ and $F(\beta)$ such that
$max(m,n) < [K:F] < n \cdot m$.
What is the minimal polynomial of $\sqrt{2}+\sqrt{3}$ over ${\mathbb Q}$
Let $f(x) = x^4 - 5$.
(a) Determine the splitting field $E$ of $f(x)$ over ${\mathbb Q}$.
(b) Compute $[E:{\mathbb Q}]$ and provide a linear basis for $E$
over ${\mathbb Q}$.
(c) Find the splitting field of $f(x)$ over ${\mathbb R}$. Compute
$[K : {\mathbb R}]$.
Let $F \subseteq E \subseteq K$ be a tower of fields such that
$K = F(\alpha)$ with $\alpha$ algebraic over $F$. Prove that
if $F \neq E$, then $m_{\alpha,F}(x) \in F[x]$ is not irreducible
in $E[x]$.
In each case below a field $F$ and a polynomial $f(x) \in F[x]$ are given.
Either prove that $f$ is irreducible over $F$ or factor $f(x)$ into
irreducible polynomials in $F[x]$. Find $[K:F]$, where $K$ is a splitting
field for $f$ over $F$.
(a) $F = {\mathbb Q}(\sqrt{-3})$, $f(x) = x^3-3$
(b) $F = {\mathbb Q}$, $f(x) = x^3 - x^2 - 5x + 5$
(2020/3/24)
Find the lattice of fields of the splitting field of $x^4-5$ over ${\mathbb Q}$.
