Algebra (Math 200 B)

Winter 2024

 Lectures: T, Th 9:30 AM--10:50 PM APM 5402 Office Hour: T, Th 11:00 AM--12:00 AM APM 7230
 TA: Unfortunately no one! ( ucsd edu) Office hour: TBA TBA

Book

Here is a list of good textbooks that you can consult for reference. I will be posting my lecture notes in this page, and I hope you would find them useful. I will not cover the topics in the same order as they are presented in the main text book; and we will learn some topics that are not covered in the main book. The main book has an excellent collection of problems and examples; and I encourage you to go over them and ask me or Francois if you have difficulty solving any of them.

• Dummit and Foote, Abstract Algebra third edition. (The main text book).
• Isaacs, Algebra .
• Hungerford, Algebra .
• Morandi, Field and Galois theory . (For the second half of the course).

Topics

This is a continuation of math 200 a. In this course we continue the study of rings and then move to module theory. The second half of the course is devoted to field and Galois theory. Our goal is to cover all the main topics covered in the first 14 chapters of Dummit and Foote's text.

Assignments.

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

• Due Jan 20: Here is the problem set.
• Due Jan 27: Here is the problem set.
• Due Feb 3: Here is the problem set.
• Due Feb 10: Here is the problem set.
• Due Feb 18: Here is the problem set.
• Due Feb 25: Here is the problem set.
• Due Mar 3: Here is the problem set.
• Due Mar 10: Here is the problem set.
• Not due: Here is the problem set.
My notes.

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

Since some parts of field theory will be covered in 200C by Professor Bucur, I am summarizing the precise topics that I covered during the lectures. You can review them from my notes or your book.
• Suppose $$E/F$$ is a field extension and $$\alpha\in E$$ is algebraic over $$F$$. Then the following statements hold:
• There is a unique monic polynomial $$m_{\alpha,F}(x)\in F[x]$$ such that for all $$f\in F[x]$$, $$f(\alpha)=0$$ if and only if $$m_{\alpha,F}|f$$. The polynomial $$m_{\alpha,F}$$ is called the minimal polynomial of $$\alpha$$ over $$F$$.
• The minimal polynomial $$m_{\alpha,F}$$ is irredcuible in $$F[x]$$; moreover if $$p(x)$$ is irreducible in $$F[x]$$ and $$p(\alpha)=0$$, then $$p(x)=cm_{\alpha,F}(x)$$ for some $$c\in F$$.
• $$F[\alpha]\simeq F[x]/\langle m_{\alpha,F}\rangle$$, $$F[\alpha]$$ is a field, and $$(1,\alpha,\ldots,\alpha^{d-1})$$ is an $$F$$-basis of $$F[\alpha]$$, where $$d=\deg m_{\alpha,F}$$. In particular, $$[F[\alpha]:F]=\deg m_{\alpha,F}$$.
• For every irreducible polynomial $$f\in F[x]$$, there exists a pair $$(E,\alpha)$$ such that
• $$E/F$$ is a field extension and $$E=F[\alpha]$$.
• $$f(\alpha)=0$$.
• Suppose $$\theta:F\to F'$$ is a field isomorphism and $$f\in F[x]$$ is irreducible. Suppose $$(E,\alpha)$$ and $$(E',\alpha')$$ satisfy the following properties: $$E/F, E'/F'$$ are field extensions, $$E=F[\alpha], E'=F'[\alpha']$$, and $$f(\alpha)=f^\theta(\alpha)=0$$. Then there is a $$\theta$$-isomorphism $$\widehat{\theta}:E\to E'$$ such that $$\widehat{\theta}(\alpha)=\alpha'$$.
• For every polynomial $$f\in F[x]$$, there exists a splitting field $$E$$ of $$f$$ over $$F$$.
• Suppose $$\theta:F\to F'$$ is a field isomorphism, $$f\in F[x]$$, $$E$$ is a splitting field of $$f$$ over $$F$$, and $$E'$$ is a splitting field of $$f^\theta$$ over $$F'$$. Then there exists an isomorphism $$\widehat{\theta}:E\to E'$$ which is an extension of $$\theta$$. In particular, a splitting field of $$f$$ over $$F$$ is unique up to an $$F$$-isomorphism.
• For every prime $$p$$ and positive integer $$n$$, there is a unique up to an isomorphism field of order $$p^n$$. It is denoted by $$\mathbb{F}_{p^n}$$. Field $$\mathbb{F}_{p^n}$$ is a splitting field of $$x^{p^n}-x$$ over $$\mathbb{Z}/p\mathbb{Z}$$. We have $$x^{p^n}-x=\prod_{\alpha\in \mathbb{F}_{p^n}} (x-\alpha).$$
• A polynomial $$f\in F[x]$$ is called separable if $$f$$ does not have multiple zeros in a splitting field of $$f$$ over $$F$$. We proved that $$f$$ is separable if and only if $$\gcd(f,f')=1$$. An irreducible polynomial $$f$$ is separable if and only if $$f'\neq 0$$; in particular, over a field of characteristic zero, every irreducible polynomial is separable. If $$F$$ is a field of characteristic $$p>0$$, then for every $$f\in F[x]$$, there exists a separable polynomial $$g\in F[x]$$ such that $$f(x)=g(x^{p^k})$$ for some non-negative integer $$k$$.
• Suppose $$K$$ is an intermediate field of $$E/F$$. Then $$[E:F]=[E:K][K:F]$$.
• Suppose $$\theta:F\to F'$$ is a field isomorphism, $$f\in F[x]$$, $$E$$ is a splitting field of $$f$$ over $$F$$, and $$E'$$ is a splitting field of $$f^\theta$$ over $$F'$$. Then $$|{\rm Isom}_{\theta}(E,E')|\leq [E:F]$$ and the equality holds precisely when all the irredcuible factors of $$f$$ in $$F[x]$$ are separable. In particular, if $$E$$ is a splitting field of $$f$$ over $$F$$, then $$|{\rm Aut}(E/F)|\leq [E:F]$$ and equality holds precisely when all the irredcuible factors of $$f$$ in $$F[x]$$ are separable.
• Suppose $$F$$ and $$F'$$ are two subfields of $$E$$, $$\theta:F\to F'$$ is an isomorphism, and $$[E:F]< \infty$$. Then $$|{\rm Isom}_\theta(E,E)|\leq [E:F]$$.
• An algebraic extension $$E/F$$ is called
• a normal extension if for all $$\alpha\in E$$, $$m_{\alpha,F}$$ factors into linear terms over $$E$$,
• a separable extension if for all $$\alpha\in E$$, $$m_{\alpha,F}$$ is separable.
• a Galois extension if it is both normal and separable.
• Suppose $$E/F$$ is a finite extension. Then the following statements are equivalent.
• There exists a polynomial $$f\in F[x]$$ with separable irreducible factors such that $$E$$ is a splitting field of $$f$$ over $$F$$.
• $$|{\rm Aut}(E/F)|=[E:F]$$.
• $$E/F$$ is a Galois extension.
• Suppose $$G$$ is a subgroup of the group of automorphisms of a field $$E$$. Consider the action of $$G$$ on $$E^n$$. Suppose $$V$$ is a non-zero subspace of $$E^n$$ which is $$G$$-invariant. Then $$V^G$$ is non-zero.
• Suppose $$G$$ is a subgroup of the group of automorphisms of a field $$E$$. Then $${\rm Fix}(G)$$ is a subfield of $$E$$ and $$G\subseteq {\rm Aut}(E/{\rm Fix}(G))$$.
• Suppose $$G$$ is a finite subgroup of the group of automorphisms of a field $$E$$. Then $$[E:{\rm Fix}(G)]\leq |G|$$.
• Prove that $${\rm Aut}_{\mathbb{F}_p}(\mathbb{F}_{p^n})=\langle \sigma_p\rangle$$ where $$\sigma_p:\mathbb{F}_{p^n}\to \mathbb{F}_{p^n}, \sigma_p(a):=a^p$$ is the Frobenius endomorphism.
• Suppose $$E/F$$ is a Galois extension and $$K$$ is an intermediate subfield of $$E/F$$. Then $$E/K$$ is a Galois extension.
• Suppose $$E/F$$ is a finite Galois extension and $$K$$ is an intermediate subfield of $$E/F$$. Then the following properties are equivalent:
• $$K/F$$ is a normal extension.
• For every $$\theta\in {\rm Aut}_F(E)$$, $$\theta(K)=K$$.
• Suppose $$E/F$$ is a finite Galois extension and $$K$$ is an intermediate subfield of $$E/F$$ such that $$K/F$$ is a normal extension. Then the restriction induces a group homomorphism $$r_{E/K}:{\rm Aut}_F(E)\to {\rm Aut}_F(K)$$ and the following is a SES: $$1\to {\rm Aut}_K(E)\to {\rm Aut}_F(E) \xrightarrow{r_{E/K}}{\rm Aut}_F(K)\to 1.$$
• When $$E/F$$ is a Galois extension, we write $${\rm Gal}(E/F)$$ instead of $${\rm Aut}_F(E)$$.
• The fundamental theorem of Galois theory. Suppose $$E/F$$ is a finite Galois extension. Let $${\rm Int}(E/F)$$ be the set of all the intermediate subfields of $$E/F$$ And $${\rm Sub}({\rm Gal}(E/F))$$ be the set of all the subgroups of $${\rm Gal}(E/F)$$. Let $$\Phi:{\rm Int}(E/F)\to {\rm Sub}({\rm Gal}(E/F)), \Phi(K):={\rm Gal}(E/K),$$ and $$\Psi:{\rm Sub}({\rm Gal}(E/F)) \to {\rm Int}(E/F), \Psi(G):={\rm Fix}(G).$$ Then the following statements hold.
• $$\Phi$$ and $$\Psi$$ are inverse of each other; this means $${\rm Gal}(E/{\rm Fix}(G))=G$$ and $${\rm Fix}({\rm Gal}(E/K))=K$$.
• $$\Phi(K)$$ is a normal subgroup of $${\rm Gal}(E/F)$$ if and only if $$K/F$$ is a normal extension. Consequently, $$\Psi(G)/F$$ is a normal extension if and only if $$G$$ is a normal subgroup $${\rm Gal}(E/F)$$.
• $$\Phi$$ and $$\Psi$$ are reverse ordering isomorphisms between the POSets $${\rm Int}(E/F)$$ and $${\rm Sub}({\rm Gal}(E/F))$$.
• Suppose $$F$$ is a field with an element $$\zeta$$ of multiplicative order $$n$$. Suppose either the charactersitic of $$F$$ is zero or it does not divide $$n$$. Suppose $$a\in F^\times$$. Let $$E$$ be a splitting field of $$x^n-a$$ over $$F$$. Then the following statements hold.
• $$E/F$$ is a finite Galois extension and $$E=F[\sqrt[n]{a}]$$ where $$\sqrt[n]{a}\in E$$ is a zero of $$x^n-a$$.
• $$x^n-a=\prod_{i=0}^{n-1}(x-\zeta^i \sqrt[n]{a})$$.
• $$\iota: {\rm Gal}(E/F)\to \langle \zeta\rangle,\quad \iota(\theta):=\frac{\theta(\sqrt[n]{a})}{\sqrt[n]{a}}$$ is an injective group homomorphism. In particular, $${\rm Gal}(E/F)$$ is a cyclic group and its order divides $$n$$.
Homework

• Homework are due on Saturdays 9:00 pm. You should post them in GradeScope.
• Late Homework are not accepted.
• There will be 9 problem sets. Your cumulative homework grade will be based on the best 8 of the 9.
• You can work on the problems with your classmates and discuss them with me, 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.
• 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).
• A good portion of the exams will be based on the weekly problem sets. So it is extremely important for you to make sure that you understand each one of them.

• Your weighted score is the best of
• Homework 25%+ midterm 25%+ Final 50%
• Homework 25%+ Final 75%
• Your letter grade is determined by your weighted score. And it is meant to suggest how your current performance corresponds to your likely result on the qualifying exam to be held next year: A = PhD Pass, A- = Provisional PhD Pass, B+/B = Masterâ€™s Pass, B- or less = not likely to pass the qual.
Further information
• There is no make-up exam.
• No notes, textbooks, calculators and electronic devices are allowed during exams.
Exams.

• Midterm:
• Time: Feb 15, 2024.
• Location: this is an in-class exam.
• Topics: All the topics that are discussed in class, till the end of Lecture on Feb 10.
• Questions are fairly similar to the homework assignments and the examples discussed in the class. Make sure that you know how to solve all of them.
• The final exam:
• Date: 03/19/2024, Tu, 8-11am.
• Location: In class.
• Topics: All the topics that were discussed in class, your homework assignments, and relevant examples and exercises in your book.
• Questions are fairly similar to the homework assignments and the examples discussed in the class. Make sure that you know how to solve anyone of them.