Here is a list of good textbooks that you can consult for reference. I will post summary of the lectures here, you can use my notes from 2018 or 2019. First we continue studying field theory, then we will be following Atiyah-Macdonald rather closely. Finally we will be using other sources for the dimension theory part (I will follow my old lecture notes).

• Morandi, Field and Galois theory . (For field theory).
• My lecture notes when I taught the 100 series.
• Atiyah and Macdonald, Introduction to commutative algebra. (The main text book).
• Matsumura, Commutative ring theory. (More advanced than needed for this course; more like a referenece).
• Eisenbud, Commutative algebra with a view toward algebraic geometry. (More advanced than needed, but it is a good book to browse through. It has some computational aspects as well.).

This is a continuation of math 200 a-b. In this course, after discussing Galois theory, we will learn about commutative algebra. These topics are mostly needed for algebraic number theory and algebraic geometry. But at the same time the materials give students a kind of vision that can be helpful in other subjects of mathematics. Topics from the first three chapters of the main book are part of the algebra qualification examination.

Problem sets will be posted here. Make sure to refresh your bowser.

• Due April 11: Here is the first problem set.
• Due April 25: Here is the second problem set.
• Due April 25: Here is the third problem set.
• Due May 2: Here is the fourth problem set.
• Due May 9: Here is the fifth problem set.
• Due May 16: No HW due. Most of the quals are in this week. Algebra qual is on May 14.
• Due May 30: Here is the sixth problem set.
• Due May 30: Here is the seventh problem set.
• Due June 8: (Optional) Here is the eighth problem set.

You can use my lecture notes from (1) , (2) 2017-2018, and (3) .

• 4/1. (1) Recalled some of the main results that we proved in field theory as part of math200b. Defined a perfect field, and proved that every polynomial over a perfect field is irreducible-separable. Proved the following major theorem:

  suppose \(E\) is a splitting field of \(f\) over \(F\), \(F\xrightarrow{\theta}F'\) is an isomorphism, \(E'\) is a splitting field of \(\theta(f)\) over \(F'\). Then

  • \(|{\rm Isom}_\theta(E,E')|\leq [E:F]\)
  • we have \(|{\rm Isom}_\theta(E,E')|= [E:F]\) if and only if \(f\) is irreducible-separable over \(F\).
  Deduce that if \(E\) is a splitting field of \(f\) over \(F\), then \(|{\rm Aut}(E/F)|\leq [E:F]\) and equality holds if and only if \(f\) is irreducible-separable over \(F\). Showed that if \(F,F'\) are two subfields of \(E\) and \(F\xrightarrow{\theta}F'\) is an isomorphism, then \(|{\rm Aut}_\theta(E)|\leq [E:F]\). Defined a normal, separable, and Galois extension. Proved that if \(F,F'\) are two subfields of \(E\), \(F\xrightarrow{\theta}F'\) is an isomorphism, and \(|{\rm Aut}_\theta(E)|= [E:F]\), then \(E/F\) is a Galois extension. Proved most part of the following major theorem: suppose \(E/F\) is a finite extension. Then the following statements are equivalent.
  • \(E\) is a splitting field of an irreducible-separable polynomial \(f\in F[x]\) over \(F\).
  • \(|{\rm Aut}(E/F)|=[E:F]\).
  • \(E/F\) is a Galois extension.
• 4/3. (2) Finished the proof of three equivalent criteria to be a finite Galois extension. Proved that \(E\) is a splitting field of a polynomial over \(F\) if and only if \(E/F\) is a finite normal extension. Along the way, we proved that if \(F\subseteq E\subseteq L\) is a tower of field extensions then the following holds
  • assuming \(E/F\) is a finite normal extension, the restriction is a well-defined group homomorphism \(r_{LE}:{\rm Aut}(L/F)\to {\rm Aut}(E/F)\).
  • assuming \(E/F\) and \(L/F\) are finite normal extensions, the following is a SES \(1\to {\rm Aut}(L/E)\to {\rm Aut}(L/F)\to {\rm Aut}(E/F)\to 1\).
  Next, we proved that if \(E/F\) is a finite normal extension, then \({\rm Aut}(E/F)\)-orbit of \(\alpha\in E\) is the set of zeros of \(m_{\alpha}\) in \(E\). Deduce that the set of fixed points of \({\rm Aut}(E/F)\) in \(E\) is \(\{\alpha\in E\mid \exists d, m_\alpha(x)=(x-\alpha)^d\}\). In particular, if \(E/F\) is a finite Galois extension, then \({\rm Fix}({\rm Aut}(E/F))=F\). To prove the converse of this statement. We showed the following technical lemma: suppose \(G\) is a finite subgroup of \({\rm Aut}(E)\). Let \(G\) act on the vector space \(E^n\). Suppose \(V\subseteq E^n\) is a subspace which is \(G\)--invariant and non-zero. Then the set of \(G\)-fixed points \(V^G\) is non-zero. (This is part of a theory known as Galois descent). Used this to show that if \(G\) is a finite subgroup of \({\rm Aut}(E)\), then \({\rm Fix}(G)\) is a subfield of \(E\) and \(E/{\rm Fix}(G)\) is a Galois extension, and \({\rm Aut}(E/{\rm Fix}(G))=G\).
• 4/8. (3) Proved Galois' fundamental theorem: Suppose \(E/F\) is a finite Galois extension. let \({\rm Int}(E/F)\) be the poset of intermediate subfields of \(E/F\) and \({\rm Sub}({\rm Gal}(E/F))\) be the poset of subgroups of \({\rm Gal}(E/F)\). Let \(\mathcal{G}:{\rm Int}(E/F)\to {\rm Sub}({\rm Gal}(E/F))\) be \(\mathcal{G}(K):={\rm Gal}(E/K)\), and \(\mathcal{F}:{\rm Sub}({\rm Gal}(E/F))\to {\rm Int}(E/F)\) be \(\mathcal{F}(G):={\rm Fix}(G)\). We argued why they are well-defined and inverse of each other. Then discussed that they are order-reversing. Next, we showed that they induce bijections between \(K/F\) normal and normal subgroups of \({\rm Gal}(E/F)\). Defined \(\inf\) and \(\sup\) in a poset, and explored some of the consequences of Galois' fundamental theorem. Looked at the example of \(\mathbb{F}_{p^n}/\mathbb{F}_p\). Saw, how Galois' fundamental theorem shows that the following is an isomorphism of posets from the poset of positive divisors of \(n\) to the poset of subfields of \(\mathbb{F}_{p^n}\): \(d\mapsto \mathbb{F}_{p^d}\). Hence, for \(d_1,d_2|n\), \(\mathbb{F}_{p^{d_1}}\cap \mathbb{F}_{p^{d_2}}=\mathbb{F}_{p^{\gcd(d_1,d_2)}}\) and \(\mathbb{F}_{p^{d_1}}\mathbb{F}_{p^{d_2}}=\mathbb{F}_{p^{{\rm lcm}(d_1,d_2)}}\). Next, we studied cyclic Kummer extensions: if \(F\) has enough roots of unity, then the splitting field of \(x^n-a\) over \(F\) is a cyclic extension of \(F\).
• 4/10. No lecture.
• 4/15. (4) Finished the study of cyclic Kummer extensions. Studied cyclotomic extensions. Showed that their Galois group is always a subgroup of \((\mathbb{Z}/n\mathbb{Z})^\times\). Defined the \(n\)-th cyclotomic polynomial over a field \(F\) assuming the characteristic of \(F\) does not divide \(n\). Showed that over \(F\), \(x^n-1=\prod_{d|n} \Phi_{d,F}(x)\). Next, we showed that \(\Phi_{n,F}(x)\in Z[x]\) where \(Z\) is the subring of \(F\) which is generated by the unity of \(F\). Showed that \({\rm Gal}(F[\zeta_n]/F)\simeq \mathbb{Z}/n\mathbb{Z})^\times\) if and only if \(\Phi_{n,F}(x)\) is irreducible over \(F\). Proved that \(\Phi_n\) is irreducible over \(\mathbb{Q}\). Defined solvable-by-radical extension and polynomials. Gave the outline of the argument, why every solvable-by-radical extension is contain in a Galois extension with solvable Galois group.
• 4/18. (5) Proved that if \(\chi_1,\ldots,\chi_m:G\to E^\times\) are distinct group homomorphisms. Then they are \(E\)-linearly independent. Suppose \(E/F\) is a finite cyclic extension and \({\rm Gal}(E/F)=\langle \sigma_0\rangle\). For \(a\in E^\times\), let \(\tau_a:E\to E, \quad \tau_a(e):=a\sigma_0(e)\). Then we showed that the minimal polynomial of \(\tau_a\) is equal to its characteristic polynomial and it is \(x^n-N_{E/F}(a)\). Along the way, we defined what the norm of an element is assuming \(E/F\) is a finite Galois extension, and showed that \(N_{E/F}:E^\times \to F^\times\) is a group homomorphism. Next, we proved that if \(E/F\) is a cyclic extension, then there exists \(\alpha\in E\) such that the Galois conjugates of \(\alpha\) form an \(F\)-basis of \(E\). After that, we proved Hilbert's theorem 90: suppose \({\rm Gal}(E/F)=\langle \sigma_0\rangle\). Then \(N_{E/F}(a)=1\) if and only if there exists \(b\in E^\times\) such that \(a=\frac{\sigma_0(b)}{b}\). This result is used to show that if \({\rm Gal}(E/F)\simeq \mathbb{Z}/n\mathbb{Z}\) and \(F\) has an element of order \(n\), then \(E=F[\sqrt[n]{a}]\) for some \(a\in F^\times\). We finished the lecture by proving Galois's theorem: suppose \(F\) is a field of characteristic zero and \({\rm Gal}(E/F)\) is a finite solvable group. Then \(E/F\) is solvable-by-radicals.
• 4/22. (6) Defined an algebraically closed field and proved some of its basic properties. Defined an algebraic closure of a field. Proved the existence of an algebraic closure of a field. Proved that if \(L/E\) and \(E/F\) are algebraic extensions, then \(L/F\) is an algebraic extension. Defined the algebraic closure of \(F\) in \(E\), and proved that it is the largest intermediate subfield of \(E/F\) which is algebraic over \(F\). Proved that if a field extension is generated by algebraic elements, then the extension is algebraic. Proved that if \(\overline{F}_i\) is an algebraic closure of \(F_i\) and \(\sigma:F_1\to F_2\) is a field isomorphism, then there exists a \(\sigma\)-isomorphism \(\widehat{\sigma}:\overline{F}_1\to \overline{F}_2\). In particular, there is a unique algebraic closure of \(F\) up to an \(F\)-isomorphism.
• 4/24. (7) Proved that if \(\overline{F}\) is an algebraic closure of \(F\) and \(E/F\) is an algebraic extension, then there exists an \(F\)-embedding of \(E\) into \(\overline{F}\). Proved that if \(E\in {\rm Int}(\overline{F}/F)\) and \(E/F\) is a normal extension, then the following is a SES \(1\to {\rm Aut}(\overline{F}/E)\to {\rm Aut}(\overline{F}/F)\to {\rm Aut}(E/F)\to 1\). Next, we discussed that \(\overline{F}\) is the union of \(E\)'s where \(E\) ranges over the normal finite extensions of \(F\) that are subfields of \(\overline{F}\). We argued that if \(E_1,E_2\in {\rm Int}(\overline{F}/F)\) and \(E_i/F\) is normal, then \(E_1E_2/F\) is normal. Then, we proved that \( {\rm Aut}(\overline{F}/F)\simeq \varprojlim {\rm Aut}(E/F) \) as \(E\in {\rm Int}(\overline{F}/F)\) ranges over finite normal extensions of \(F\). Next, we recalled extension and contraction of ideals. Recalled that contraction of a prime ideal is a prime ideal. Showed \(\mathfrak{a}^{\rm ece}=\mathfrak{a}^{\rm e}\) and \(\mathfrak{b}^{\rm cec}=\mathfrak{b}^c\). Deduced that the extension and the contraction maps induce bijections between the sets of contracted and extended ideals. Defined the Zariski-topology on \({\rm Spec}(A)\), and proved that \(f^{\ast}:{\rm Spec}(B)\to {\rm Spec}(A)\) is continuous. We finished the lecture by showing that \(\pi^\ast\) gives us a closed immersion from \({\rm Spec}(A/\mathfrak{a})\) to \({\rm Spec}(A)\) and its image is the closed set \(V(\mathfrak{a})\) of prime divisors of \(\mathfrak{a}\).
• 4/29. (8) Explored the extension and the contraction maps induced from \(A\to S^{-1}A, a\mapsto \frac{a}{1}\). Mentioned that \(\mathfrak{a}^{e}=S^{-1}\mathfrak{a}\) and so every ideal of \(S^{-1}A\) is an extended ideal. Proved that \(\mathfrak{a}^{{\rm ec}}=\bigcup_{s\in S} (\mathfrak{a}:s)\) and deduce that an ideal \(\mathfrak{a}\) is a contradcted ideal if and only if for every \(s\in S\), \(\mathfrak{a}=(\mathfrak{a}:s)\), which is equivalent to saying that \(\{s+\mathfrak{a}\mid s\in S\}\cap D(A/\mathfrak{a})=\varnothing\). Used this to deduce that the contraction and the extesion maps give us bijections between \(\{\mathfrak{p}\in {\rm Spec}(A)\mid \mathfrak{p}\cap S=\varnothing\}\) and \({\rm Spec}(S^{-1}A)\). In particular, there is a bijection between \({\rm Spec}(A_{\mathfrak{p}})\) and \(\{\mathfrak{p}'\in {\rm Spec}(A)\mid \mathfrak{p}'\subseteq \mathfrak{p}\}\). Next, we defined a primary ideal. Proved that \(\mathfrak{q}\) is primary if and only if \({\rm Nil}(A/\mathfrak{q})=D(A/\mathfrak{a})\). Showed that if \(\mathfrak{q}\) is primary, then \(\sqrt{\mathfrak{q}}\) is prime. Mentioned that the converse does not hold. But if \(\sqrt{\mathfrak{q}}\) is a maximal ideal, then \({\mathfrak{q}}\) is primary. In particular, if \({\mathfrak{m}}\) is maximal, then \({\mathfrak{m}}^n\) is \({\mathfrak{m}}\)-primary for every positive integer \(n\). Used this to show that in a PID, an ideal is primary if and only if it is a power of a prime ideal. Defined \((\mathfrak{a}:\mathfrak{b})\) and mentioned some of its basic properties. Showed that in a PID, \((\langle a\rangle:\langle b\rangle)=\langle a/\gcd(a,b) \rangle\). Defined decomposable ideals. Proved that if \(\mathfrak{q}_1\) and \(\mathfrak{q}_2\) are two \(\mathfrak{p}\)-primary ideals, then \(\mathfrak{q}_1\cap \mathfrak{q}_2\) is also \(\mathfrak{p}\)-primary. Used this to show that a decomposable ideal has a reduced primary decomposition.
• 5/1. (9) Proved that if \(\mathfrak{q}\) is \(\mathfrak{p}\)-primary, then \((\mathfrak{q}:x)= \begin{cases} A & \text{if } x\in \mathfrak{q} \\ \mathfrak{q} & \text{ if } x\not\in \mathfrak{p} \\ \mathfrak{p}-\text{primary} & \text{ if } x\not\in \mathfrak{q}. \end{cases}\) (The first uniqueness) Proved that if \(\mathfrak{a}=\bigcap_{i=1}^n \mathfrak{q}_i\) is a reduced primary decomposition and \(\mathfrak{p}_i=\sqrt{\mathfrak{q}_i}\), Then \(\{\mathfrak{p}_1,\ldots,\mathfrak{p}_n\}={\rm Spec}(A)\cap \{\sqrt{(\mathfrak{a}:x)}\mid x\in A\}\). In particular, \(\{\mathfrak{p}_1,\ldots,\mathfrak{p}_n\}\) does not depend on the choice of a reduced primary decomposition. \(\mathfrak{p}_i\)'s are called primes associated with \(\mathfrak{a}\), and \(\{\mathfrak{p}_1,\ldots,\mathfrak{p}_n\}\) is denoted by \({\rm Ass}(\mathfrak{a})\). Next, we proved that for every \(\mathfrak{p}\in V(\mathfrak{a})\), there exists \(\mathfrak{p}'\in {\rm Ass}(\mathfrak{a})\) such that \(\mathfrak{p}'\subseteq \mathfrak{p}\). Deduce that the set of minimal elements of \(V(\mathfrak{a})\) is the same as the set of minimal elements of \({\rm Ass}(\mathfrak{a})\). Proved that if \(\mathfrak{a}\) is decomposable, then \(\{x\in A\mid (\mathfrak{a}:x)\neq \mathfrak{a}\}=\{x\in A\mid x+\mathfrak{a}\in D(A/\mathfrak{a})\}=\bigcup_{\mathfrak{p}\in {\rm Ass}(\mathfrak{a})} \mathfrak{p}\). Deduce that if \(0\) is decomposable, then \(A\) has only finitely many minimal prime ideals, \(D(A)=\bigcup_{\mathfrak{p}\in {\rm Ass}(0)} \mathfrak{p}\), and \({\rm Nil}(A)=\bigcap_{\mathfrak{p}\in {\rm Ass}(0)} \mathfrak{p}\). Next, we said \(S(\mathfrak{a}):=\mathfrak{a}^{{\rm ec}}\) with respect to the natural map \(A\to S^{-1}A\). Proved that \(S^{-1}\mathfrak{q}=S^{-1}A\) and \(S(\mathfrak{q})=A\) if \(\mathfrak{q}\) is \(\mathfrak{p}\)-primary and \(\mathfrak{p}\cap S\neq \varnothing\). Next, we proved that \(S(\mathfrak{q})=\mathfrak{q}\) if \(\mathfrak{q}\) is \(\mathfrak{p}\)-primary and \(\mathfrak{p}\cap S= \varnothing\). We, also proved that \(\sqrt{S^{-1}\mathfrak{q}}=S^{-1}\mathfrak{p}\) if \(\mathfrak{q}\) is \(\mathfrak{p}\)-primary.
• 5/6. (10) Recalled that \(S(\mathfrak{a}):=\mathfrak{a}^{\rm ec}=\bigcup_{s\in S} (\mathfrak{a}:s)\) with respect to the ring homomorphism \(A\to S^{-1}A, a\to \frac{a}{1}\). Deduce that \(\sqrt{\mathfrak{a}}\cap S\neq \varnothing\) implies that \(S^{-1}\mathfrak{a}=S^{-1}A\) and \(S(\mathfrak{a})=A\). Next, we proved that if \(\mathfrak{q}\) is \(\mathfrak{p}\)-primary and \(S\cap \mathfrak{p}=\varnothing\), then \(S(\mathfrak{q})=\mathfrak{q}\), \(S^{-1}\mathfrak{q}\) is \(S^{-1}\mathfrak{p}\)-primary. We also pointed out that if \(\widetilde{q}\) is a \(\widetilde{p}\)-primary ideal of \(S^{-1}A\), then \(\widetilde{q}^{\rm c}\) is a \(\widetilde{p}^{\rm c}\)-primary and \(\widetilde{q}=\widetilde{q}^{\rm ce}\) and \(\widetilde{p}=\widetilde{p}^{\rm ce}\). Using these results, we showed how a reduced primary decomposition behaves under localization. Next, we defined an isolated subset of \({\rm Ass}(\mathfrak{a})\) and proved the second uniqueness theorem: if \(\mathfrak{a}=\bigcap_{i=1}^n \mathfrak{q}_i\) is a reduced primary decomposition and \(\sqrt{\mathfrak{q}_i}=\mathfrak{p}_i\) and \(\Sigma\subseteq {\rm Ass}(\mathfrak{a})\) is an isolated subset, then \(\bigcap_{\mathfrak{p}_i\in \Sigma} \mathfrak{q}_i\) depends only on \(\mathfrak{a}\). In particular, if \(\mathfrak{p}_i\) is a minimal element of \({\rm Ass}(\mathfrak{a})\), then \(\mathfrak{q}_i\) depends only on \(\mathfrak{a}\). Used this to show that if \(D\) is an integral domain of dimension 1, then a decomposable ideal has a unique primary decomposition. Next, we start working towards the existence of a primary decomposition. We defined an irreducible ideal. Proved that in a Noetherian ring every proper ideal can be written as an intersection of finitely many irreducible ideals. Finished the lecture by proving that every irreducible ideal in a Noetherian ring is primary.
• 5/8. (11) Proved that in a Noetherian ring, every proper ideal has a primary decomposition. Deduce that in a Noetherian ring there are only finitely many minimal prime ideals. Hence, there is a finite subset of \({\rm Spec}(A)\) which is dense. Then, we defined an integeral extension and an integeral ring homomorphism. For a ring extension \(B/A\), we proved that the following statements are equivalent:
  • \(b\in B\) is integral over \(A\).
  • \(A[b]\) is a finitely generated \(A\)-module.
  • There exists an intermidiate ring \(A[b]\subseteq C\subseteq B\) such that \(C\) is a finitely generated \(A\)-module.
  • There exists a faithful \(A[b]\)-module \(M\) that is a finitely generated \(A\)-module.
  Proved that the integral closure of \(A\) in \(B\) is a subring of \(B\). Showed that if \(B/A\) and \(C/B\) are integral extensions, then \(C/A\) is an integral extension. Deduce that the integral closure of \(A\) in \(B\) is integrally closed in \(B\). Next, we defined integrally closed integral domains. Proved that a UFD is integrally closed. Pointed out that if \(B/A\) is an integral extension, then \(A/\mathfrak{b}^{\rm c}\to B/\mathfrak{b}\) and \(S^{-1}A\to S^{-1}B\) are integral. Finished the lecture by proving that an integral domain \(D\) is integrally closed if and only if \(D_{\mathfrak{p}}\) is integrally closed for every \(\mathfrak{p}\in {\rm Spec}(D)\) if and only if \(D_{\mathfrak{m}}\) is integrally closed for every \(\mathfrak{m}\in {\rm Max}(D)\). Pointed out that this means being integrally closed is a local property. Mentioned that this concepts has geometric meaning and it essentially says that there are no codimension 1 singularities. In particular, for affine curves, we get smoothness.
• 5/13. (12) Proved that if \(B/A\) is an integral extension and \(B\) is an integral domain, then \(A\) is a field precisely when \(B\) is a field. Deduce that if \(f:A\to B\) is integral, then \(f^{\ast}({\rm Max}(B))\subseteq {\rm Max}(A)\). Using this result and localization at a prime ideal, we proved the following important theorem: if \(B/A\) is an integral extension, then
  • \(f^{\ast}:{\rm Spec}(B)\to {\rm Spec}(A)\) is surjective.
  • \(f^{\ast}({\rm Max}(B))={\rm Max}(A)\).
  • \(f^{\ast}(V(\mathfrak{b}))=V(\mathfrak{b}^{\rm c})\). In particular, \(f^{\ast}\) is a closed map.
  This was used to prove the Going-Up theorem. Next, we showed that the dimension of a fiber is 0 if \(B/A\) is an integral extension. Deduce that if \(B/A\) is an integral extension, then \(\dim A=\dim B\). As a consequence, we showed that the ring of integers of a number field is integrally closed of dimension 1. Defined being integral over an ideal of the base ring.
• 5/15. (13) Assuming that \(B\) is an integral domain, \(B/A\) is a ring extension, and \(C\) is the integral closure of \(A\) in \(B\) and \(\mathfrak{a}\) is an ideal of \(A\), we proved that \(b\in B\) is integral over \(A\) if and only if \(b\in \sqrt{\mathfrak{a}^{\rm e}}\). Next, we showed that if \(B/A\) is an integral extension, \(B\) is an integral domain, \(A\) is integrally closed, \(F\) is a field of fractions of \(A\), \(\mathfrak{a}\) is an ideal of \(A\), and \(b\) is integral over \(\mathfrak{a}\), then the minimal polynomial \(m_{b,F}(x)=x^d+a_{d-1}x^{d-1}+\cdots+a_0\) for some \(a_i\in \sqrt{\mathfrak{a}}\). Next, we proved the Going-down theorem. As a lemma, we showed part of your HW assignment: a prime ideal is in the image of \(f^{\ast}:{\rm Spec}(B)\to {\rm Spec}(A)\) if and only if it is a contracted ideal.
• 5/20. (14) We gave the outline of a second proof for the Going-Down theorem. Used the Going-Down theorem to prove that \(f^{\ast}:{\rm Spec}(B)\to {\rm Spec}(A)\) is an open map if \(A\) is integrally closed, \(B/A\) is an integral extension and \(B\) is an integral domain. Next, we defined a valuation ring. Mentioned that if \(D\) is a UFD and \(p\) is an irreducible element, then \(D_{\langle p\rangle}\) is a valuation ring. Proved that if \(A\) is a valuation ring and \(F\) is its field of fractions, then (1) \(A\) is a local ring, (2) if \(A\subseteq A'\), then \(A'\) is a valuation ring, and (3) \(A\) is integrally closed. Then, we stated the following important technical theorem: suppose \(F\) is a field, \(A\) is a subring of \(F\), \(\Omega\) is an algebraically closed field, and \(\theta_0:A\to \Omega\) is a ring homomorphism. Let \( \Sigma:=\{(B,\theta)\mid A\subseteq B\subseteq F, \theta|_A=\theta_0\}. \) Then (1) \(\Sigma\) has a maximal element, (2) if \((B,\theta)\) is a maximal element of \(\Sigma\), then \(B\) is a local ring and \({\rm Max}(B)=\{\ker \theta\}\), and (3) \(B\) is a valuation and \(Q(B)=F\). We finished the proof of the first two parts, and mentioned our plan for proving the third part.
• 5/22. (15) We finished proof of the techincal theorem. Used it to show that the integral closure of an integral domain \(A\) in its field of fractions \(F\) is equal to the intersection of all the valuation subrings of \(F\) which contain \(A\). Mentioned that this result is important for the construction of an abstract curve. Next, we proved Chevalley's theorem: suppose \(A\) is an integral domain, \(B\) is a finitely generated \(A\)-algebra, \(A\subseteq B\), \(B\) is an integral domian. Then for every \(b_0\in B\setminus \{0\}\), there exists \(a_0\in A\setminus \{0\}\) such that the following holds. If \(\Omega\) is an algebraically closed field, \(\theta:A\to \Omega\) is a ring homomorphism, and \(\theta(a_0)\neq 0\), then there exists \(\widehat{\theta}:B\to \Omega\) such that \(\widehat{\theta}|_A=\theta\) and \(\widehat{\theta}(b_0)\neq 0\). Before proving this theorem, we explained its connection with the representable functors in the catergory of unital commutative rings. Haveing a ring homomorphism \(A\xrightarrow{f} B\), we get a natural transformation \(h_B\xrightarrow{f^\ast} h_A\). We would like to understand the image of \(f^{\ast}(h_B(\Omega))\). The mentioned theorem says that for every \(b_0\in B\setminus \{0\}\) there exists \(a_0\in A\setminus \{0\}\) such that \(f\) extends to \(A_{a_0}\to B_{b_0}\), and \(f^{\ast}:h_{B_{b_0}}(\Omega)\to h_{A_{a_0}}(\Omega)\) is surjective. We recalled that the functor \(V_{\mathfrak{a}}\) from the category of \(k\)-algebras to the category of sets. Here, \(\mathfrak{a}\) is an ideal of \(k[x_1,\ldots,x_n]\) and \(V_{\mathfrak{a}}(R):=\{(r_1,\ldots,r_n)\in R^n\mid \forall f\in \mathfrak{a}, f(r_1,\ldots,r_n)=0\}\). We recalled that there is a natural bijection between \(V_{\mathfrak{a}}\) and the representable functor \(h_A\) where \(A:=k[x_1,\ldots,x_n]/\mathfrak{a}\). In particular, if \(B\) is a finitely generated \(\Omega\) algebra and \(B\) is an integral domain and \(\Omega\) is an algebraically closed field, then \(h_{B_{b_0}}(\Omega)\to h_{\Omega}(\Omega)\) is surjective implies that \(h_B(\Omega)\neq 0\). Hence, we deduce a version of Nullstellensatz. (This will be explained later with more details). Next, we proved Chevalley's theorem.
• 5/27. (16) Proved four verions of Hilbert's Nullstellensatz. Stated two versions of Noether's normalization. Proved one implies the other one. Defined the GKdim of a finitely generated \(k\)-algebra. Gave an outline of an argument on why GKdim A= dim A if A is a finitely generated unital commutative k-algebra (no detail was provided, and the argument relied on the fact that \(\dim k[x_1,\ldots,x_n]=n\)).
• 5/29. (17) Discussed automorphisms of the ring of polynomials. Proved Noether's normalization lemma. Mentioned general ideas of elimination process. Defined the height of a prime ideal. Mentioned that \({\rm ht}(\mathfrak{p})=\dim A_{\mathfrak{p}}\). Argued why \(\dim A\geq {\rm ht}(\mathfrak{p})+\dim A/\mathfrak{p}\). Start proof of the following theorem: if \(A\) is a finitely generated \(k\)-algebra, then \(\dim A= {\rm ht}(\mathfrak{p})+\dim A/\mathfrak{p}\).
• 6/3. (18) Proved that if \(A\) is a finitely generated \(k\)-algebra, then \(\dim A= {\rm ht}(\mathfrak{p})+\dim A/\mathfrak{p}\) for every \(\mathfrak{p}\in {\rm Spec}(A)\). Defined Artinian modules and a composition series of a module. Proved that every two composition series have the same length and every chain of submodules is part of a composition series. Deduce that \(\ell(M)=\ell(N)+\ell(M/N)\). Show that \(M\) is Noetherian and Artinian if and only if \(\ell(M)\) is finite. Proved that if \(A\) is Artinian, then \({\rm Spec}(A)\) is a finite discrete set. Showed that \({\rm Nil}(A)\) is nilpotent if \(A\) is Artinian. Next, we showed that \(A\) is Artinian if and only if \(A\) is a Noetherian dimension zero ring.
• 6/5. (19) Proved that for a local Artinian ring \(A\) the following are equivalent: (1) Every ideal is principal, (2) The maximal ideal is principal, (3) \(\dim_{A/\mathfrak{m}}(\mathfrak{m}/\mathfrak{m}^2)\leq 1\). Proved various equivalent conditions for a ring to be a DVR. Proved various equivalent conditions for a ring to be a Dedekind domain. Deduced that in a Dedekind domain every non-zero proper ideal has a unique factorization into maximal ideals. We ended the lectures by stating Krull's PIT and HT, and giving a geoemtric interpatation for them. Stay curious!

• Homework are due on Fridays 9:00 pm unless it is specified for another time. You should post them in GradeScope.
• Late Homework are not accepted.
• There will be 8 problem sets. Your cumulative homework grade will be based on the best 5 of the 7.
• You can work on the problems with your classmates and discuss them with me or Hugo, but you have to write down your own version. Copying from other's solutions is not accepted and is considered cheating. Copying from an online solution bank is not acceptable, either. Using AIs is not allowed, either. But after you submit your solutions, you can try them and let me know if the AIs can solve the problems.
• Only selected problems will be scored, but you are responsible for understanding all the posted problems (including the ones that are not part of the homework assignments).

• Your grade is based on your performance in your weekly homework assignments.