(2019/1/8)
Prove that if coproducts exists in a category,
they are unique up to isomorphism.
Prove that if products exist in a category, then coproducts exist
in the dual category.
Give an example of a category where the product does not exist.
More specifically
prove that the category Field does not have a product.
A second category that we discussed in class was that of a pre-order
$P$ which is not a lattice (does not have a notion of gcd) is does not have a product.
A good example of poset which is not a lattice
is the poset with two elements which are not comparable.
(2019/1/10)
Give an example of a non-concrete category (resolve the confusion:
is the forgetful functor and the example
of the category with one object and two morphisms such an example or not? give
a more interesting one).
Reference an explanation on a blog
from 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.
Is the category of $\mathbb N$ a concrete category?
Show that $F(X) = \{ f : X \rightarrow R ~| ~f$ has finite support $\}$
are free objects in the category of R-Mod.
Show that if $( {\mathcal C}, {\mathcal U} )$ is a concrete category
and $F$ is an functor from $F : $ Set $\rightarrow {\mathcal C}$
mapping a set $X$ in Set to the free objects $F[X]$ in ${\mathcal C}$,
prove that if $X \simeq X'$, then $F[X] \simeq F[X']$.
Note: a message from Nantel says that this problem is wrong.
Define functor $F : $ Set $\rightarrow$ Grp that defines
free groups in the category of Grp and show
that it satisfies the defintion of being free objects in that category.
Prove the isomorphism theorems for R-Mod.
(2019/1/15)
Prove that an arbitrary direct sum of modules is again a module and the direct
sum is a submodule.
[DF, Ch. 10.3, #20 a, direct sum = direct product with finite support]
Give an example of where the direct sum is not isomorphic to the direct product.
[DF, Ch. 10.3, #20 b]
Show that the direct sum of free R-modules is again free. [DF, Ch. 10, #23]
Prove that an arbitrary direct product of free modules need not be free.
[DF, Ch. 10.3, #24]
Prove that free modules over non-commutative rings need to have a unique
rank. [DF, Ch. 10.3, #27]
(2019/1/17)
If I is a left ideal of $R$ and $M$ is a left $R$ module, let
$$I M = \{ \sum_{i} a_i m_i | a_i \in I, m_i \in M \}$$
where the sums are finite. Show that $IM$ is a submodule of $M$.
[DF, Ch 10.1, #5]
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]
Let $I$ be a left ideal of $R$ and let $n$ be a positive integer. Prove
$$R^n/IR^n \simeq (R/IR) \times \cdots \times (R/IR)~~~~~n\hbox{ times }$$
[DF, Ch 10.2, #12]
Assume that $R$ is commutative. Prove that $R^n \simeq R^m$ if and only
if $n=m$ (i.e. two free $R$ modules of finite rank are isomorphic if and
only if they have the same rank).
Note: if $R$ is commutative then there is a maximal ideal $I$ of $R$.
(see for instance
http://www.math.uconn.edu/~kconrad/blurbs/ringtheory/ideals.pdf
Theorem 6.7) and hence $R/I$ is a field.
Note: you can assume (and reduce the problem to) the equivalent
statement for fields is true, that is, $F^n \simeq F^m$ if and only if
$n=m$ [DF, Ch. 11.1]
(2019/1/29)
For any ring $R$, that $M$ is finitely generated if and only if there is a surjective
$R$-homomorphism $\phi : R^n \rightarrow M$ for some integer $n$. [DF, Ch. 12.1 #16]
The book outlines a slightly different exposition of the proof I am presenting
in class, but see exercises. [DF, Ch. 12.1 #17-#19]
(2019/2/5)
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]
Let $R$ be a PID and $N$ an $R$-module
$$N \simeq \bigoplus_{i=1}^r R/ p^{\alpha_i} R$$
for some prime $p$ of $R$ and $1 \leq \alpha_1 \leq \alpha_2 \leq \cdots \alpha_r$.
Prove that $N/ pN \simeq F^r$ where
$F = R/(p)$. [DF, 12.1, Theorem 9]
Let $R$ be a PID and $N$ an $R$-module and assume that
$$N \simeq \bigoplus_{i=1}^r R/ p^{\alpha_i} R$$
where $\alpha_1 = \alpha_2 = \cdots = \alpha_s =1$, $\alpha_s>1$ and
$1 \leq \alpha_1 \leq \alpha_2 \leq \cdots \alpha_r$.
Show that $N/pN \simeq \bigoplus_{i=s+1}^r R/p^{\alpha_i-1} R$. [DF, 12.1, Theorem 9]
(2019/2/14)
Recall, a module $P$ is projective if every short exact sequence
$0 \rightarrow A \rightarrow M \rightarrow P \rightarrow 0$ splits
($M \simeq A \oplus P$).
Show that $P$ is projective iff for all $M$ such that there exists a
surjection $\phi: M \rightarrow P$, there is a $P' \leq M$ such that
$M = P' \oplus ker~\phi$. (detail)
Recall, a module $Q$ is injective if every short exact sequence
$0 \rightarrow Q \rightarrow M \rightarrow B \rightarrow 0$ splits
($M \simeq Q \oplus B$).
Show that $P$ is injective iff for all $M$ such that there exists an
injection $\psi: Q \rightarrow M$, there is a $N' \leq M$ such that
$M = \psi(Q) \oplus N'$.
If $A \oplus B$ is injective, then $A$ is injective. [DF, sec 10.5 #4]
(2019/2/25)
Prove that if $P$ and $P'$ are projective then $P \oplus P'$ is projective.
(solution) [DF, sec 10.5 #3]
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. [DF, p.403, sec 10.5 # 1 (a)]
Let $A$ be the free abelian group generated by $\langle a_1, a_2,
a_3, a_4\rangle$. Let $B$ be the subgroup generated by
$B=\langle 3a_1-a_2-a_3, -a_1+3a_2-a_3, -a_1-a_2+3a_3\rangle$
(a) Find a basis $\langle x_1, x_2, x_3, x_4\rangle$ of $A$
and a generating set $\langle y_1, y_2, \ldots, y_s \rangle$ of $B$ such that
$s\le 4$, $y_i=\alpha_i x_i$ and $\alpha_1|\alpha_2|\ldots|\alpha_s$.
(b)
Describe a cannonical form for $A/B$ explicitly.
What are the Betti number and invariant factors?
(2019/3/12)
Note that $x^3-2 = (x - \sqrt[3]{2})(x^2 + \sqrt[3]{2} x + \sqrt[3]{4})$.
Prove that $x^2 + \sqrt[3]{2} x + \sqrt[3]{4}$ is is irreducible over
${\mathbb Q}( \sqrt[3]{2})$. [D&F p.537 Ex (3)]
Prove that $x^3-3x-1$ is irreducible over ${\mathbb Q}$
[see Proposition 11 of Ch. 9]
and that $\sqrt{2}$ is not contained in
${\mathbb Q}(\alpha)$ where $\alpha$ is a root of $x^3-3x-1$.
[D&F Example 5, p.521]
Prove that the minimal polynomial of $\sqrt[6]{2}$ over
${\mathbb Q}(\sqrt{2})$ is $x^3 - \sqrt{2}$
(that is, show that $x^3-\sqrt{2}$ is irreducible over ${\mathbb Q}(\sqrt{2})$
[D&F Example 2, p. 524 and Example 3 p. 521]
Determine the degree of $2 + \sqrt{3}$ and $1+\sqrt[3]{2}+\sqrt[3]{4}$
over ${\mathbb Q}$. [D&F 13.2 Ex. 4 p. 530]
(2019/3/14)
Give an example of a polynomial over a field which is irreducible,
but not separable. [D&F Example 13.5 (2) p. 546]
Determine the splitting field and degree over ${\mathbb Q}$ for
each of $x^4-2$, $x^4+2$, $x^4 + x^2 + 1$, $x^6-4$. [D&F Sec 13.4, Exercises 1-4]
Let $K_1$ and $K_2$ be finite extensions of $F$ contained in the field $K$,
and assume that both are splitting fields over $F$.
Prove that their composite $K_1 K_2$ is a splitting field over $F$.
Let $\sqrt{3 + 4i}$ denote the square root of the complex number $3+4i$ that
lies in the first quadrant and let the $\sqrt{3 - 4i}$ denote the square root of $3-4i$
denote the square root of $3 - 4i$ that lies in the fourth quadrant.
Prove that $[{\mathbb Q}(\sqrt{3 + 4i} + \sqrt{3 - 4i}) : {\mathbb Q} ] = 1$.
[Section 13.2 Exercise 11 (a)]
Determine whether the following polynomials are irreducible over the given field.
(as a review of irreducibility criteria in section 9.4 which creeps into
results in these sections)
[Section 9.4 Exercises 1,2]
(a) $x^2 + x + 1$ in ${\mathbb F}_2[x]$
(b) $x^3 + x + 1$ in ${\mathbb F}_3[x]$
(c) $x^4 + 1$ in ${\mathbb F}_5[x]$
(d) $x^4 + 10x^2 + 1$ in ${\mathbb Z}[x]$
(e) $x^4 - 4x^3 + 6$ in ${\mathbb Z}[x]$
(f) $x^6+30x^5-15x^3+6x-120$ in ${\mathbb Z}[x]$
(g) $x^4+4x^3+6x^2+2x+1$ in ${\mathbb Z}[x]$
(h) $\frac{(x+2)^p - 2^p}{x}$ where $p$ is an odd prime in ${\mathbb Z}[x]$
Show that $[{\mathbb Q}(\sqrt[3]{2}, \sqrt{-3}):{\mathbb Q}] = 6$.
Find the minimal polynomial of $\sqrt[3]{2} - \sqrt{-3}$. [D&F, Sec 13.4, Example (3) p. 537]
