- Lecture 0: Here is a quick review of the main results and concepts in field that we have already discussed.
- Lecture 1: Here we more or less proved the main theorem of Galois theory. It will be stated in the next lecture.
- Lecture 2: Here we proved:
main theorem of Galois theory, \(E/F\) is a simple extension if and only if there are only finitely many intermediate subfields, if \(E/F\) is a finite separable extension, then there are only finitely many intermediate subfields, a finite separable field extension is a simple extension, \(\overline{F}/F\) is Galois if and only if \(\overline{F}/F\) is separable if and only if any algebraic extension \(E/F\) is separable, extending field of coefficients does not change \(\gcd(f,g)\).
- Lecture 3: Here we proved:
an effective criterion for a polynomial to be separable; (perfect fields) for a field \(F\), we have \(F^p=F\) if and only if any algebraic extension \(E/F\) is separable (here \(p\) is 1 if characteristic of \(F\) is zero and it is the characteristic of \(F\) otherwise); various properties of \({\rm Gal}(\overline{\mathbb{F}}_p/\mathbb{F})\); \({\rm Gal}(F[\zeta_n]/F)\) can be embedded into \((\mathbb{Z}/n\mathbb{Z})^{\times}\); defined the \(n\)-th cyclotomic polynomial.
- Lecture 4: Here we proved and mentioned:
Cyclotomic polynomials are irreducible in \(\mathbb{Q}[x]\); solvability by radicals; Galois's theorem; \({\rm Gal}(\mathbb{Q}[\zeta_n]/\mathbb{Q})\simeq (\mathbb{Z}/n\mathbb{Z})^{\times}\); Dedekind's linear independence of characters; Hilbert's theorem 90; and mentioned the connection between Hilbert's theorem 90 and Galois's theorem.
- Lecture 5: Here we proved and mentioned:
Galois's theorem on solvability of polynomials; defined Jacobson radical; defined the set of prime divisors of an ideal; almost proved that considering the set of prime divisors of ideals as closed subsets of \({\rm Spec}(A)\) we get a topology which is called Zariski topology.
- Lecture 6: Here we proved and mentioned:
a result on finite union of prime ideals; \({\rm Max}(A)\) intersects any non-empty closed set; closed points of \({\rm Spec}(A)\) are precisely elements of \({\rm Max}(A)\); 0 is dense in \({\rm Spec}(A)\) if \(A\) is an integral domain; contraction and extension of ideals; \(\mathfrak{a}^{ece}=\mathfrak{a}^e\) and \(\mathfrak{b}^{cec}=\mathfrak{b}^c\); if \(f:A\rightarrow B\) is a ring homomorphism, then contraction gives us a continuous map \(f^*:{\rm Spec}(B)\rightarrow {\rm Spec}(A)\); \(\pi:A\rightarrow A/\mathfrak{a}\) induces a homeomorphism \(\pi^*:{\rm Spec}(A/\mathfrak{a})\rightarrow V(\mathfrak{a})\); for \(f:A\rightarrow S^{-1}A\) an ideal of \(S^{-1}A\) is an extended ideal; \(f^*\) gives us a bijection between \({\rm Spec}(S^{-1}A)\) and \(\{\mathfrak{p}\in {\rm Spec}(A)| \mathfrak{p}\cap S=\varnothing\}\).
- Lecture 7: Here we proved and mentioned:
how to understand \({\rm Spec}(S^{-1}A)\); how to understand the fiber \((f^*)^{-1}(\mathfrak{p})\) where \(f:A\rightarrow B\) is a ring homomorphism; \(\mathfrak{p}\) is in the image of \(f^*\) if and only if \(\mathfrak{p}^{ec}=\mathfrak{p}\); generalizations of Nakayama's lemma for commutative rings; primary ideals and their basic properties.
- Lecture 8: Here we proved and mentioned:
\(\mathfrak{m}\)-primary ideals where \(\mathfrak{m}\) is a maximal ideal; description of primary ideals in a PID; quotient ideal of a primary ideal; reduced primary decomposition; 1st uniqueness of reduced primary decompositions and primes associated with a decomposable ideal; union of primes associated with 0 is the set of zero-divisors.
- Lecture 9: Here we proved and mentioned:
while union of primes associated with 0 is the set of zero-divisors, their intersection is the set of nilpotent elements; extension-contraction under localization: for any ideal and for primary ideals; localization and reduced primary decomposition; 2nd uniqueness of reduced primary decompositions; definition of Krull dimension and consequence of the 2nd uniqueness theorem for one dimensional integral domains.
- Lecture 10: Here we proved and mentioned:
irreducible ideals; in a Noetherian ring irreducible implies primary; in a Noetherian ring any proper ideal is decomposable; integral extension; various equivalent conditions of integrality; integral closure in a ring extension; tower of integral extensions; integral closure of \(A\) in \(B\) is integrally closed in \(B\); a UFD is integrally closed.
- Lecture 11: Here we proved and mentioned:
relation between integrality, going to a quotient ring, and localization; localization of the integral closure of \(A\) in \(B\) is the integral closure of \(S^{-1}A\) in \(S^{-1}B\); being integrally closed is a local property; if \(f:A\hookrightarrow B\) is integral, then \(f^*:{\rm Spec}(B)\rightarrow {\rm Spec}(A)\) is surjective and closed, \(f^*({\rm Max}(B))={\rm Max}(A)\) and \((f^*)^{-1}({\rm Max}(A))={\rm Max}(B)\), \(f^*(V(\mathfrak{b}))=V(\mathfrak{b}^c)\); and Going-Up Theorem.
- Lecture 12: Here we proved and mentioned:
fibers of an integral morphism have dimension 0; \(B/A\) integral implies \(\dim A=\dim B\); \(b\in B\) is integral over \(\mathfrak{a}\) if and only if \(b\in \sqrt{\mathfrak{a}^e}\); if \(A\) is integrally closed and \(b\in B\) is integral over \(\mathfrak{a}\), then the minimal polynomial of \(b\) over the field of fractions of \(A\) has non-leading coefficients in \(\sqrt{\mathfrak{a}}\); Going-Down Theorem.
- Lecture 13: Here we proved and mentioned:
- Lecture 14: Here we proved and mentioned:
- Lecture 15: We proved and mentioned:
Based on a technical theorem on valuation rings, we proved many results including four versions of Hilbert's Nullstellensatz.
- Lecture 16: We proved and mentioned:
Using Hilbert's Nullstellensatz, we proved a finitely generated \(k\)-algebra that is an integral domain is a Jacobson ring. Then we proves Noether's normalization. Next we started dimension theory by focusing on dimension zero Noetherian rings. To get a better understanding of such rings, we studied finite length modules.
- Lecture 17: We proved and mentioned:
We proved many properties of submodule series of a finite length modules. A module has finite length if and only if it is both Noetherian and Artinian. Artinian rings have dimension zero and discrete spec. The Jacobson radical of an Artinian ring is nilpotent. A ring is Artinian if and only if it is a zero-dimensional Noetherian ring. Any Artinian ring is a unique direct product of local Artinian rings.
- Lecture 18: We proved and mentioned:
We proved a local Artinian ring is a principal ideal ring if and only if \(\dim_{A/\mathfrak{m}}\mathfrak{m}/\mathfrak{m}^2\le 1\). We proved various equivalent forms of saying a ring is a Discrete Valuation Ring. We proved equivalent properties of saying a ring is a Dedekind domain. Proved that if the height of the ideal generated by a is zero, then a is a zero-divisor.
- Lecture 19: We proved and mentioned:
Proved Krull's Principal ideal theorem, Krull's Height Theorem, \(\dim k[x_1,\ldots,x_n]=n\), a converse of Krull's Height Theorem, dimension of a Noetherian local ring with maximal ideal \(\mathfrak{m}\) is equal to the minimum of \(d(\mathfrak{q})\) as \(\mathfrak{q}\) ranges in \(\mathfrak{m}\)-primary ideals.