| My notes.
You can use my lecture notes from 2018 and
2019.
I will summarize what gets covered during the lectures here. You should go over the notes and your book on the relevant topics.
- 1/7.
Recalled prime ideals, maximal ideals, set of prime divisors of an ideal, prime element, irreducible element, being associates. Defined a Euclidean Domain. Proved that \(\mathbb{Z}\), \(F[x]\) where \(F\) is a
field, \(\mathbb{Z}[i]\) are ED. Proved that ED implies PID. Proved that in every integral domain prime element is irreducible. Showed that in a PID irreducible implies prime. Defined a UFD. Showed that in a UFD
every irredicble is prime.
- 1/9.
Proved that if every non-zero non-unit element in an integral domain \(D\) can be written as a product of irreducibles, then \(D\) is a UFD precisely when every irreducible is prime. Defined Noetherian rings.
Proved that the following are equivalent: (1) \(A\) is Noetherian. (2) \(A\) satisfies the ascending chain condition (acc). (3) Every ideal of \(A\) is finitely generated. Proved that every non-empty collection of
ideals of a Noetherian ring has a maximal element. Proved that in a Noetherian integral domain, every non-zero non-unit element can be written as a product of irreducibles. Deduced that PID implies UFD. Defined the dimension
of a ring. Proved that \(\dim A=0\) if and only if \({\rm Spec}(A)={\rm Max}(A)\). Deduced that if \(D\) is an integral domain, then \(\dim D=0\) if and only if \(D\) is a field. Proved that if \(D\) is a PID, then \(\dim D\leq 1\).
This is equivalent to saying that \({\rm Spec}(D)={\rm Max}(D)\cup \{0\}\). Proved that \(\dim A[x]\geq \dim A+1\). Showed that \(A[x]\) is a PID if and only if \(A\) is a field.
Stated Hilbert's basis theorem and started its proof.
- 1/14. Proved Hilbert's basis theorem. Deduced that \(A[x_1,\ldots,x_n]\) is Noetherian if \(A\) is Noetherian, \(F[x_1,\ldots,x_n]\) is Noetherian if \(F\) is a field,
and \(\mathbb{Z}[x_1,\ldots,x_n]\) is Noetherian. Proved that quotient of a Noetherian ring is Noetherian. Argued why a finitely generated ring is a quoteint of \(\mathbb{Z}[x_1,\ldots,x_n]\),
and deduced that a finitely generated ring is Noetherian. Next, we stated that if \(D\) is a UFD, then \(D[x]\) is a UFD, and started the steps towards its proof. In a UFD \(D\),
we fixed a set \(\mathcal{P}\) of representatives from the classes of associates of its irreducible elements. Then, being a UFD implies that there exist functions
\(v_p:D\setminus \{0\}\to \mathbb{Z}^{\geq 0}\) and \({\rm sgn}:D\setminus \{0\}\to D^\times\) such that for all \(a\in D\setminus \{0\}\), \(a={\rm sgn}(a)\prod_{p\in \mathcal{P}} p^{v_p(a)}\). Argued why
\(v_p\) is a monoid homomorphism and \(0\to D^\times \to (D\setminus \{0\})\to \bigoplus_{p\in \mathcal{P}}\mathbb{Z}^{\geq 0}\to 0\) is a short exact sequence of monoids, where
\(a\mapsto (v_p(a))_{p\in \mathcal{P}}\). Showed that \(a|b\) if and only if \(v_p(a)\leq v_p(b)\) for all \(p\). Defined the gcd of elements in a UFD. Pointed out that
\(\gcd(ca_1,\ldots,ca_n)=|c|\gcd(a_1,\ldots,a_n)\), and if \(\gcd(a_1,\ldots,a_n)=d\), then \(\gcd(a_1/d,\ldots,a_n/d)=1\). Defined primitive polynomials and proved the first verison of Gauss's lemma: product of two
primitive polynomials is primitive. Defined the content of a polynomial, and proved the second version of Gauss's lemma: content of product is the product of contents. Recalled what the field of fractions of and
integeral domain is and mentioned its universal property.
- 1/16. Proved the third version of Gauss's lemma. Deduced that for a primitive polynomial of positive degree, irreducibility over the field of fractions is equivalent to irreducibility over the initial UFD.
We also proved that if \(f\) is primitive and \(g\in D[x]\), then \(f|g\) in over \(D\) if and only if \(f|g\) over the field of fractions of \(D\). Next, we showed for \(d\in D\), \(d\) is irreducible in \(D\) if and only if
it is irreducible in \(D[x]\). Put all the pieces together and proved that if \(D\) is a UFD, then \(D[x]\) is a UFD. Next, we used Gauss's lemma to prove mod \(\mathfrak{a}\) irredcubility criterion. Stated Eisenstein's
irreduciblity criterion.
- 1/21. Completed the proof of Eisenstein's irreducibility criterion. Defined what a module is. Mentioned that modules over fields are exactly vector spaces. Mentioned that submodules of a ring are exactly its ideals.
Hence, module theory combines ideal theory with linear algebra. Defined induced module structure. Mentioned \(A^n\) can be viewed as an \({\rm M}_n(A)\), and using induced modules, for every \(a\in {\rm M}_n(A)\), we can view
\(A^n\) as an \(A[x]\)-module. This module is denoted by \(V_a\). Started parametrizing all the possible \(A\)-module structures on an abelian group \(M\): the following maps are bijections and are inverse of each other,
\(
{\rm Mod}(A,M)\to {\rm Hom}(A, {\rm End}(M)),
\quad
c(a,m):=a\cdot m \text{ is sent to } a\mapsto \ell_a \text{ where } \ell_a(m):=a\cdot m.
\)
and
\(
{\rm Hom}(A, {\rm End}(M))\to {\rm Mod}(A,M),\quad
f \text{ is sent to the induced module structure}.
\)
We also proved that \({\rm End}(M)\) is a ring and \(M\) is a left \({\rm End}(M)\)-module.
- 1/23. Completed the proof of the mentioned parametrization of \(A\)-module structures on an abelian group \(M\). Defined the direct product and the external direct sum of a family of modules.
Introduced the concept of a category. Mentioned what a functor is, and pointed out what a representable functor is. Vaguely mentioned what a natural transformation is (we will return to this later). Proved that there
is a natural bijection between the functors \(h_{\bigoplus_{i\in I} M_i}\) and \(\prod_{i\in I}h_{M_i}\), and there is a natural bijection between the contravariant functors \(h^{\prod_{i\in I} M_i}\) and
\(\prod_{i\in I} h^{M_i}\). These means there are natural bijections \(\eta_N:{\rm Hom}_A(\bigoplus_{i\in I} M_i,N)\to \prod_{i\in I}{\rm Hom}_A(M_i,N)\) and
\(\eta_N: {\rm Hom}_A(N, \prod_{i\in I} M_i)\to \prod_{i\in I} {\rm Hom}_A(N, M_i)\). We introduced the free \(A\)-module generated by a set \(X\).
- 1/28. Completed the proof of the universal property of a free \(A\)-module. Defined and proved properties of internal direct sum of submodules. Defined a quotient module.
Proved the 1st, the 2nd, and the 3rd isomorphism theorems for modules. Mnetioned the bijection theorem for submodules of a quotient module. Proved that
\(\frac{\bigoplus_{i\in I}M_i}{\bigoplus_{i\in I}N_i}\simeq \bigoplus_{i\in I}\frac{M_i}{N_i}\). Defined the annihilator of a module. Proved that an \(A\)-module \(M\) can be viewed as an \(A/{\rm Ann}(M)\)-module and
the set of \(A\)-submodules is the same as the set of \(A/{\rm Ann}(M)\)-submodules. For an ideal \(\mathfrak{a}\) and an \(A\)-module, we defined \(\mathfrak{a}M\) and proved that it is a submodule of \(M\). Showed that
\(\mathfrak{a}A^n=\mathfrak{a}^n\). Proved that if \(f:M\to N\) is an \(A\)-module homomorphism, then \(f(\mathfrak{a}M)\subseteq \mathfrak{a}N\).
Argued that \(\mathfrak{a}\subseteq {\rm Ann}(M/\mathfrak{a}M)\),
and so, we can view \(M/\mathfrak{a}M\) as an \(A/\mathfrak{a}\)-module. Proved that if \(f:M\to N\) is an \(A\)-module homomorphism, then
\(\overline{f}:M/\mathfrak{a}M\to N/\mathfrak{a}N, \quad \overline{f}(x+\mathfrak{a}M):=f(x)+\mathfrak{a}N\) is a well-defined \(A/\mathfrak{a}\)-module homomorphism.
In particular, for every maximal ideal \(\mathfrak{m}\),
we get an \(A/\mathfrak{m}\)-linear map between the vector spaces \(M/\mathfrak{m}M\) and \(N/\mathfrak{m}N\). Used this to show that if \(f:A^n\to A^m\) is surjective, then \(n\geq m\). Defined \(A\)-linearly independent elements
of an \(A\)-module, \({\rm rank}_A(M)\), and the minimium number \(d_A(M)\) of generators of \(M\). In your HW assignment, you showed that \({\rm rank}_A(M)\leq d_A(M)\) and eqaulity holds precisely when \(M\) is a free \(A\)-module
(assuming \(M\) is finitely generated). In class, we present a much simpler proof to show \({\rm rank}(D^n)=n\) if \(D\) is an integral domain.
- 1/30. Proved the classification of submodules of \(D^n\) when \(D\) is a PID. Argued why such a statement holds only when \(D\) is a PID. Proved the existence part of the
funtamental theorem of finitely generated modules over a PID.
- 2/4. Proved the generalized Chinese Remainder Theorem (for an arbitrary unital commutative ring). Proved the uniqueness part of the fundamental theorem of finitely generated modules over a PID. Defined the invariant
factors of a matrix over a field.
- 2/6. Proved the existence and uniqueness of rational canonical forms. Proved the Cayley-Hamilton theorem and showed that irreducible factors of the characteristic polynomial of a matrix divide its minimal polynomial.
Worked towards the Jordan form of a matrix.
- 2/11. Proved the existence and the uniqueness of the Jordan forms. Pointed out how to find the invariant factors knowing the Jordan form of a matrix. Mentioned the relation between the invariant factors \(a\) and
the minimum number of generators of the \(F[x]\)-module \(V_a\). Defined the Short Exact Sequence (SES) of modules and their homomorphism. Proved that every SES is isomorphic to a standard SES.
- 2/13. Proved the Short Five Lemma and various properties of a splitting SES. (Then we had our midterm).
- 2/18. Enriched representable functors in the category of \(A\)-modules to a functor from \(A\)-modules to \(A\)-modules. Proved that \(h_M\) is left exact. Showed that the folloing properties are equivalent:
(1) \(h_M\) is exact. (2) If \(f:N\to N'\) is surjective, then \(h_M(f)\) is surjective. (3) If \(f:N\to N'\) is surjective, then every \(g:M\to N'\) has a lift \(\widehat{g}:M\to N\). (4) Every SES of the form
\(0\to N\to N'\to M\to 0\) splits. (5) \(M\) is a direct summand of a free module. A module \(M\) with these properties is called a projective module. If \(D\) is an integeral domain, projective implies torsion-free.
For a finitely generated module over a PID, the following are equivalent: (1) projective (2) torsion-free (3) free. Next, we proved that over a Noetherian integral domain \(D\), an ideal \(\mathfrak{a}\) is projective
if and only if there exists a fractional ideal \(M\) such that \(\mathfrak{a}M=D\). We pointed out that an ideal is a free module if and only if it is principal. Using this you can find projective modules that are not free.
- 2/20.
Recalled what functors are. Mentioned that \(\underline{{\rm GL}}_n\) can be viewed as a functor from the category of unital commutative rings to the category of groups. Viewing representable functors of \(A\)-modules as a
functor to the category of \(A\)-modules, we looked that their composite, and asked if it is still a representable functor. To make sense of the question, we defined natural transformations. Defined the functor
\(b_{M_1,M_2}\) of \(A\)-bilinear maps from the category of \(A\)-modules to itself.
- 2/25.
Proved that \(h_{M_1}\circ h_{M_2}\simeq b_{M_1,M_2}\). Proved that there exists an \(A\)-module \(M_1\otimes_A M_2\), which is called the tensor product of \(M_1\) and \(M_2\) over \(A\) such that
\(h_{M_1\otimes_A M_2}\simeq b_{M_1,M_2}\simeq h_{M_2}\circ h_{M_1}\). Pointed out \(h_{M_1\otimes_A M_2}\simeq b_{M_1,M_2}\) is equivalent to the universal property of tensor products. We also discussed that
\(h_{M_1\otimes_A M_2}\simeq h_{M_2}\circ h_{M_1}\) is natural on \(M_i\)'s. Defined the tensor functor
\(T_M\), and obtained a natural isomorphism \({\rm Hom}_A(T_M(N), K)\simeq {\rm Hom}_A(N, h_M(K))\); we say \(T_M\) is a left adjoint of \(h_M\).
- 2/27.
Defined the contravariant representable functors \(h^M\). Proved that \(h^M\) is left-exact. Proved that \(N_1\to N_2\to N_3\) is exact if for all \(M\), \(h_M(N_1)\to h_M(N_2)\to h_M(N_3)\) is exact (if with respect to all
the observers we get an exact sequence, it is an exact sequence). Similarly, \(N_1\to N_2\to N_3\) is exact if for all \(M\), \(h^M(N_1)\leftarrow h^M(N_2)\leftarrow h^M(N_3)\) is exact. Used these to show that if \(\mathcal{F}\)
is a left adjoint of \(\mathcal{G}\), then \(\mathcal{F}\) is right-exact and \(\mathcal{G}\) is left-exact. Deduced that the tensor functor \(T_M\) is right-exact. Proved the following important isomorphism:
\(M\otimes_A A/\mathfrak{a}\simeq M/\mathfrak{a}M\). Used this to show that \(\mathbb{Z}/n\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Z}/m\mathbb{Z}\simeq \mathbb{Z}/\gcd(n,m)\mathbb{Z}\). We also showed that if \(P_1\) and \(P_2\) are projective,
then \(P_1\otimes_A P_2\) is projective.
- 3/4.
Mentioned a result: suppose \(\mathcal{F}_1,\mathcal{F}_2:\mathcal{C}\to \mathcal{C}\) are two functors. Consider the following functors \((M,N)\mapsto {\rm Hom}(\mathcal{F}_i(M),N)\). If these functors are isomorphic, then
\(\mathcal{F}_1\) and \(\mathcal{F}_2\) are isomorphic. This is equivalent to saying that if for all observers \(h^N\circ \mathcal{F}_1\simeq h^N\circ \mathcal{F}_2\) naturally on \(N\), then \(\mathcal{F}_1\simeq \mathcal{F}_2\).
We used this to give a clean proof of two facts for tensor functors:
\(T_M \circ T_N\simeq T_{M\otimes_A N}\) and
\(T_{\bigoplus_{i\in I}M_i}\simeq \bigoplus_{i\in I} T_{M_i}\).
Proved that \(\bigoplus_{i\in I} \mathcal{F}_i\) is an exact functor if and only if for all \(i\in I\),
\(\mathcal{F}_i\) is an exact functor. Defined a flat module. Proved that if \(M\) and \(N\) are flat , then so is
\(M\otimes_A N\). Showed that \(\bigoplus_{i\in I} M_i\) is flat if and only if for all \(i\in I\), \(M_i\) is flat. Proved that projective modules are flat. Mentioned the localization functor is an exact functor. Deduce that
\(\mathbb{Q}\) is a flat \(\mathbb{Z}\)-module, but it is not projective. Defined an \(A\)-algebra.
- 3/6.
Showed that the tensor product of two \(A\)-algebras is an \(A\)-algebra. Proved that \(A[x]/\mathfrak{a}\otimes_A B\simeq B[x]/\mathfrak{a}^e\) where \(\mathfrak{a}^e\) is the ideal of \(B[x]\)
that is generated by the image of \(\mathfrak{a}\) under the homomorphism given by \(A\)-algebra structure of \(B\). Used this to show
\(\mathbb{Q}[i]\otimes_{\mathbb{Q}} \mathbb{Q}[i]\simeq \mathbb{Q}[i]\oplus \mathbb{Q}[i]\)
as \(\mathbb{Q}\)-algebras. Next, we investigated \(\mathbb{Z}/p\mathbb{Z}\otimes_{\mathbb{Z}} \mathbb{Z}[i]\). Showed that
when \(p=2\), it has a nilpotent element (ramified prime), for \(p\equiv 1 \pmod{4}\), it is isomorphic to \(\mathbb{Z}/p\mathbb{Z}\oplus \mathbb{Z}/p\mathbb{Z}\) and for \(p\equiv 3 \pmod{4}\), it is isomorphic to
a field of order \(p^2\). Started field theory. Defined algebraic and transcendental elements. Defined field extensions, algebraic field extensions, and the degree of a field extension. Started proof of the following theorem:
if \(E/F\) is a field extension and \(\alpha\in E\) is algebraic over \(F\), then the following statements hold.
- (Minimal polynomial) There exists a unique monic (non-constant) polynomial \(m_{\alpha}\) in \(F[x]\) such that \(f(\alpha)=0\) if and only if \(m_{\alpha}|f\) in \(F[x]\).
- \(m_{\alpha}\) is irreducible in \(F[x]\).
- \(F[\alpha]\) is a field and it is isomorphic to \(F[x]/\langle m_\alpha\rangle\).
- \(1,\alpha,\ldots,\alpha^{d-1}\) is an \(F\)-basis of \(F[\alpha]\) where \(d=\deg m_{\alpha}\). In particular, \([F[\alpha]:F]=\deg m_{\alpha}\).
- 3/11.
Proved the mentioned theorem on algebraic elements.
Next, we showed that if \(p(x)\in F[x]\) is irreducible and \(p(\alpha)=0\), then \(p(x)={\rm ld}(p) m_{\alpha}(x)\). Used this to find out how to prove the following lemma: if \(p\) is irreducible in \(F[x]\),
then there exists a pair \(E,\alpha\) such that (1) \(E/F\) is a field extension, \(\alpha\in E\), and \(E=F[\alpha]\). (2) \(p(\alpha)=0\). A repeated application of this, we showed the existence of a splitting field of
\(f\in F[x]\) over \(F\). Then, we proved that \(\theta:F_1\to F_2\) is an isomorphism of fields, \(p\in F_1[x]\) is irreducible, \((E_i,\alpha_i)\) are pairs such that \(E_i=F_i[\alpha_i]\), \(E_i/F_i\) is a field extension,
\(p(\alpha_1)=0\) and \(p^\theta(\alpha_2)=0\), then there exists an isomorphism \(\widehat{\theta}:E_1\to E_2\) such that \(\widehat{\theta}|_{F_1}=\theta\). We refer to such an isomorphism as a \(\theta\)-isomorphism. We
think about them as isomorphism of extensions \(E_1/F_1\) and \(E_2/F_2\). Next, we proved the following extension of the uniqueness of splitting fields up to an isomorphism.
Suppose \(f\in F_1[x]\), \(\theta:F_1\to F_2\) is
an isomorphism of fields, \(E_1\) is a splitting field of \(f\) over \(F_1\) and \(E_2\) is a splitting field of \(f^\theta\) over \(F_2\).
Then the extensions \(E_1/F_1\) and \(E_2/F_2\) are isomorphic. Then, we defined
an \(F\)-isomorphism between two extensions of \(F\), and deduced that two splitting fields of \(f\) over \(F\) are \(F\)-isomorphic.
- 3/13.
Proved the existence and uniqueness of finite fields of order \(p^n\) where \(p\) is prime and \(n\) is a positive integer. Proved that a finite field of order \(p^n\) is a
splitting field of \(x^{p^n}-x\) over \(\mathbb{Z}/p\mathbb{Z}\). We also proved that \(x^{p^n}-x=\prod_{\alpha\in \mathbb{F}_{p^n}} (x-\alpha)\). We defined separable polynomials as
an element \(f(x)\in F[x]\) which does not have multiple zeros in its splitting field over \(F\). Showed that separability does not change under a base change.
Showed that the gcd of two polynomials over a field does not change under a base change (to another field). Used this to show that a polynomial \(f\) is separable if and only
if \(\gcd(f,f')=1\). Then showed that an irreducible polynomial is separable if and only if \(f'\neq 0\). Said a polynomial is irreducible separable if all of its irreducible
factors are separable. Deduce that every polynomial over a characteristic zero field is irreducible separable. Showed that if \(F\) is a field of characteristic \(p>0\) and \(f\)
is irreducible in \(F[x]\), then there exists a unique irreducible and separable polynomial \(f_{\rm s}\) and a non-negative integer \(k\) such that \(f(x)=f_{\rm s}(x^{p^k})\).
It was mentioned that if \(F^p=F\), then every irreducible polynomial is separable. Proved the tower formula: \([L:F]=[L:E][E:F]\); in particular, \(L/F\) is a finite extension
if and only if \(L/E\) and \(E/F\) are finite extensions.
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