Math 417 – Introduction to Abstract Algebra – Fall 2021

(section D13)

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Algebra is the study of operations, rules and procedures to solve equations. The origin of the term ‘Algebra’ seems to go back to a IX Century treaty by an Arab mathematician with the title ‘The Compendious Book on Calculation by al-jabr and al-muqabala’. The term al-jabr is used in this book to denote two procedures: (i) the sum of two positive quantities to both sides of an equation, in order to cancel negative terms and (ii) the multiplication of both sides of an equation by a positive number to cancel fractions. With the passage of time, the term al-jabr or algebra became synonymous of the general study of equations and operations on them.

Algebra is one of the pillars of Mathematics and this course gives an introduction to the basics of Algebra, including conceptual proofs of all main results.

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Lecturer:Rui Loja Fernandes
Email: ruiloja (at) illinois.edu
Office: 346 Illini Hall
Office Hours: Mondays and Fridays 1.30-2.30 pm (or by appointment);
Class meets: MWF 11:00-11:50 am, 341 Altgeld Hall;
Prerequisites: Officially either MATH 416 or one of MATH 410, MATH 415 together with one of MATH 347, MATH 348, CS 373; or consent of instructor. In practice, ability to understand and write proofs.

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In this page:

• Announcements
• Syllabus
• Textbooks
• Grading Policy and Exams
• Homework Assignments
• Emergency information for students in Mathematics courses

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Announcements:

• You can find the grades of the final exam on-line. A graph with the grades of the final is available here. Note that the grades of the best 10 homeworks, midterms and final exam is an integer less or equal to 100. Your total score (TS) for the course is obtained by the formula 1.5*HW+MT1+MT2+MT3+3*FE (maximum 750), so that homework is worth 20%, midterms 40% and final exam 40%. Your final grade is obtained from the total score by applying the following curve:
  • A+: TS>=740
  • A: 740>TS>=715
  • A-: 715>TS>=700
  • B+: 700>TS>=660
  • B: 660>TS>=600
  • C+: 600>TS>=550
  • C: 550>TS>=500
• Here is a solution of the final exam.
• The last day of instruction for this class is Friday, Dec. 6. Important annoucements:
  • There will be no office hours the week Dec 9-13. I will be available all afternoon in my office on Tuesday (Dec 17) and Wednesday (Dec 18), the days before the final.
  • The final will be held on Thursday, Dec 19, at 1.30 pm, in the regular classroom. I have made available a mock Final to practice.
  • The final grades will be available on this web page on Friday, Dec 19, by noon.
  • Students can consult their final in my office on Friday, Dec 19, at 2 pm. After that all grades will become final and cannot be changed.
• Students can view the scores of homework on-line (please note that you will have to log in).
• Here is a solution of 3rd Midterm and you can find your score here. I will not curve the midterm, but you can see how you did relative to your classmates in this plot of 3rd midterm grades.
• Here is a solution of 2nd Midterm and you can find your score here. I will not curve the midterm, but you can see how you did relative to your classmates in this plot of 2nd midterm grades.
• The drop deadline for this course is 2021. Please see this leaflet for information about droping this course.
• Here is a solution of 1st Midterm and you can find your score here. I will not curve the midterm, but you can see how you did relative to your classmates in this plot of 1st midterm grades.

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Syllabus:

Chapters 1-6 of the recommended text, covering: Fundamental theorem of arithmetic, congruences. Permutations. Groups and subgroups, homomorphisms. Group actions with applications. Polynomials. Rings, subrings, and ideals. Integral domains and fields. Roots of polynomials. Maximal ideals, construction of fields.

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Textbooks:

Recommended Textbook:

• Frederick M. Goodman, Algebra: Abstract and Concrete (Edition 2.6), SemiSimple Press Iowa City, IA. [PDF file available for download free of charge]

Other Textbooks: (first ref available on the web; last two refs on hold in the Math Library)

• Manuel Ricou and Rui L. Fernandes, Introduction to Algebra [these are my notes that cover the same material but in different order; may contain typos]
• Michael Artin, Algebra, (2nd edition) Prentice Hall, 1991. [some level as the recommended textbook with alternative approaches]
• Garrett Birkhoff and Saunders MacLane, A survey of modern algebra , (4th Edition), Macmillan, 1977. [a classic book; more advanced than the recommended textbook]

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Grading Policy and Exams

There will be weekly homework/quizzes, 3 midterms and a final exam. All exams/midterms will be closed book.

• Homework and quizzes (20% of the grade): Homework problems are to be assigned once a week. They are due the following week, at the beginning of the Monday class. No late homework will be accepted. There will be 13 homework assignments, but only the ten best grades will count and the other homework grades will be dropped. If necessary, quizzes may be offered during the semester.
• Midterms (40% of the grade): The midterms will take place in class on Friday Sep 20, Friday Oct 18 and Friday Nov 15 (the dates are subject to change).
• Final Exam (40% of the grade): You have to pass the final to pass the course. According to the non-combined final examination schedule it will take place Thursday, Dec 19, 1.30 pm, in the regular classroom.

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Homework Assignments and Sections covered so far:

• Homework #1: Goodman, Exercises 1.3.1, 1.3.2, 1.3.3, 1.4.1, 1.4.2, 1.4.3, 1.5.1, 1.5.2, 1.5.3. [Important Note: In the textbook, the only symmetries considered are rigid motions, i.e., rotations+translations].
• Homework #2: Goodman, Exercises 1.5.5, 1.5.6, 1.5.8, 1.5.9, 1.5.10, 1.7.4, 1.7.5, 1.7.6, 1.7.8, 1.7.9.
• Homework #3: Goodman, Exercises 1.6.3, 1.6.4, 1.6.7, 1.6.8, 1.6.9, 1.10.1, 1.10.2, 1.10.3, 1.10.4.
• Homework #4: Goodman, Exercises 1.10.5, 1.10.6, 2.1.2, 2.1.4, 2.1.5, 2.1.6, 2.1.7.
• Mock midterm 1 to practice.
• Homework #5: Goodman, Exercises 2.1.10, 2.1.11, 2.1.12, 2.2.2, 2.2.4, 2.2.6, 2.2.9, 2.2.11.
• Homework #6: Goodman, Exercises 2.2.14, 2.2.15, 2.2.16, 2.2.17, 2.2.19, 2.3.5, 2.3.7, 2.3.8, 2.4.5, 2.4.8, 2.4.9.
• Homework #7: Goodman, Exercises 2.4.13, 2.4.17, 2.4.18, 2.5.4, 2.5.6, 2.5.7, 2.5.8, 2.5.13, 2.5.14, 2.6.1, 2.6.3, 2.6.5.
• Homework #8: Goodman, Exercises 2.6.6, 2.7.3, 2.7.4, 2.7.6, 2.7.7, 3.1.9, 3.1.10, 3.1.13.
• Mock midterm 2 to practice.
• Homework #9: Goodman, Exercises 5.1.3, 5.1.5, 5.1.7, 5.1.8, 5.1.9, 5.1.12, 5.1.13, 5.1.14, 5.1.15.
• Homework #10: Goodman, Exercises 5.2.2, 5.2.4, 6.1.1, 6.1.4, 6.1.6, 6.1.10, 6.1.11, 6.2.1, 6.2.3, 6.2.4.
• Homework #11: Goodman, Exercises 6.2.7, 6.2.8, 6.2.9, 6.2.10, 6.2.11, 6.2.16, 6.2.19, 6.3.2, 6.3.7, 6.3.8.
• Homework #12: Goodman, Exercises 6.3.1, 6.3.3, 6.3.5, 6.3.6, 6.3.9, 6.3.10, 6.3.11.
• Mock midterm 3 to practice.
• Homework #13: Goodman, Exercises 6.4.1, 6.4.4, 6.4.7, 6.4.8, 6.4.13, 6.4.14, 6.4.16, 1.8.6, 1.8.7, 1.8.10.
• Mock Final to practice.

Sections of the book covered so far: 1.1-1.7, 1.10 (midterm 1 stops here), 2.1-2.7 (midterm 2 stops here), 3.1, 5.1, 5.2, 1.11, 6.1-6.3 (midterm 3 stops here at the Isomorphism Theorem for Rings), 6.4, 1.8, 6.5, 6.6.

(PDF files can be viewed using Adobe Acrobat Reader which can be downloaded for free from Adobe Systems for all operating systems.)

Frequently Asked Questions about Homework

1. How many homework assignments will there be?
2. How is the grades of the homework calculated?
3. How is each homework assignment graded?
4. Can I turn in late homework?
5. Can I turn in homework via e-mail?
6. What if I fall sick, or what if I decide to go to Las Vegas to get married, or… , and I cannot turn in homework on time?

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Math 417: Homework 8

Due Friday, April 16, 2021

（5分）练习5.4.2。

（8分）练习5.4.5。

（10分）练习5.4.6。 同构φ1，φ2，φ3是第261页给出的同构。

（10分）练习5.4.8。

（10分）证明以下任何形式的20阶同构酶都是直接同构产物：

（a）对于某些同态α：Z4→Aut（Z5）的Z5􏰀αZ4，或
（b）同态β的Z5􏰀β（Z2×Z2）：Z2×Z2→Aut（Z5）。

（10分）考虑由2×2矩阵组成的组G = SL（2，Z3），在Z3 = {[0]，[1]，[2]}中的行列式等于[1]。 找到G的3个Sylow子群，并写出其元素。 显示由

是G的2个Sylow子群

􏰁[0] [2]􏰂[1] [1]􏰂[1] [0]和[1] [2]

1. (5points)Exercise5.4.2.
2. (8points)Exercise5.4.5.
3. (10 points) Exercise 5.4.6. The homomorphisms φ1,φ2,φ3 are the ones given on page 261.
4. (10points)Exercise5.4.8.
5. (10points)Provethatanygroupoforder20isisomorphictoasemidirectproductofoneofthe following forms:(a) Z5 􏰀α Z4 for some homomorphism α : Z4 → Aut(Z5), or
   (b) Z5􏰀β(Z2×Z2)forsomehomomorphismβ:Z2×Z2→Aut(Z5).
6. (10 points) Consider the group G = SL(2,Z3) consisting of 2×2 matrices with entries in Z3 = {[0], [1], [2]} that have determinant equal to [1]. Find a 3-Sylow subgroup of G and write out its elements. Show that the subgroup generated by

is a 2-Sylow subgroup of G.

1. Show that $\mathbb{C}$ has a subfield isomorphic to $\mathbb{Q}(x)$.
2. Show that $\mathbb{Q}(\sqrt{2}+\sqrt{-1})$ is a splitting field of $x^{4}-4 \in \mathbb{Q}[x]$.
3. Let $a$ be an integer. Suppose that $\sqrt{a} \in \mathbb{Q}(\sqrt[3]{2}) .$ Show that $a=b^{2}$ for some $b \in \mathbb{Z}$.
4. (a) Let $F$ be the splitting field of $x^{3}+6 \in \mathbb{Q}[x] .$ Find $[F: \mathbb{Q}]$.
   (b) Let $F$ be the splitting field of $x^{3}+\overline{6} \in \mathbb{R}_{11}[x] .$ Find $[F: \mathbb{Q}]$.
5. Show that if $F$ is a field and $|F|=8$, then $F$ has only two subfields.
6. Let $F \subseteq \bar{Q}$ be a finite extension of $\mathbb{Q}$ and $p$ be a prime number. Let $K \subseteq \overline{\mathrm{Q}}$ be the splitting field of $\left.x^{p}-a \in F \mid x\right]$. Suppose that $[K: F]=p$.
   (a) Show that $F$ contains all the $p$ -th roots of unity.
   (b) Let
   $\operatorname{Aut}(K / F)={$ field isomorphisms $\sigma: K \rightarrow K$ such that $\sigma(c)=c$ for all $c \in F}$.
   Then $\operatorname{Aut}(K / F)$ is a group under composition. Let $\alpha \in K$ be a root of $x^{p}-a$. Show that the map
   $$
   \begin{aligned}
   \varphi: \operatorname{Aut}(K / F) & \longrightarrow K^{\times} \
   \sigma & \longmapsto \frac{\sigma(\alpha)}{\alpha}
   \end{aligned}
   $$
   is a group homomorphism.
   (c) Show that $\operatorname{Aut}(K / F) \cong \mathbb{Z} / p Z$.
7. Let $F$ be a field of characteristic $p$. Show that $\alpha \in \bar{F}$ is separable over $F$ iff $F(\alpha)=F\left(\alpha^{p}\right)$.

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