- Office Hour: M 3:40 pm - 4:40 pm, W 6:00 pm - 7:00 pm, T 11:00 am - 12:00 pm(on zoom)

- Email: wangjiuy at math dot duke dot edu

- Syllabus: Enrolled students should read carefully

- Textbook: Abstract Algebra, A Geometric Approach, Theodore Shifrin

- For class discussion: Piazza Page

- For online homework submission: GradeScope

- Jan 8:

Homework 1, due in class on Jan, 13

- We introduced definition of monoids, groups and rings.

- Show basic properties of groups and rings that follow from definition.

- We introduced definition of monoids, groups and rings.
- Jan 13:

Homework 2, due in class on Jan, 22

- We introduced the definitions of group homomorphism and ring homomorphism.

- We explained there exists a ring homomorphism from Z to every ring R.

- We introduced equivalence relations.

- We start to contruct the ring Z_m.

- We introduced the definitions of group homomorphism and ring homomorphism.
- Jan 15:

- We prove that Z_m is a ring.

- We introduced the definitions of zero-divisor, unit, integral domain and field.

- We prove that Z_m is a ring.
- Jan 22:

Quiz 1

Homework 3, due in class on Feb 3.

- We introduced Euclidean Algorithm.
- We prove that m,n are relatively prime if and only if there exist integers a and b such that am+bn =1.

- We prove that Z_m is a field if and only if m is a prime.

- Jan 27:

- We prove the unique factorization of integers into prime numbers.

- We showed that R[x], the set of polynomials over a ring R, is also a ring.

- We showed the division algorithm works for F[x] when F is a field.

- We prove the unique factorization of integers into prime numbers.
- Jan 29:

- We gave a comparison between Z and F[x].

- We introduced subring, kernel, ideal, principal ideals.

- We gave a comparison between Z and F[x].
- Feb 3:

Quiz 2

Homework 4, due in class on Feb 10.

- We prove that all ideals of Z and F[x] are principal ideal domain.

- We introduced ring isomorphism.

- We started to define quotient rings.

- We prove that all ideals of Z and F[x] are principal ideal domain.
- Feb 5:

- We discussed Z_m as quotient rings of integers.

- We proved fundamental homomorphism theorems.

- We discussed Z_m as quotient rings of integers.
- Feb 10:

Homework 5, due in class on Feb 17.

- We introduced direct product of rings.

- We proved Chinese Remainder Theorem, for both Z and general rings.

- We introduced direct product of rings.
- Feb 12:

- We prove that F=Q[x]/< f(x) > is a field when f(x) is irreducible.

- We proved F is isomorphic to Q[a] where a is a root of f(x).

- We introduced field extension.

- We prove that F=Q[x]/< f(x) > is a field when f(x) is irreducible.
- Feb 17:

- We prove that Q[\alpha_1] is isomorphic to Q[\alpha_2] when \alpha_1 and \alpha_2 are the two roots of the same polynomial.

- We prove that Q[\sqrt{2}] is isomorphic to Q[\sqrt{5}] as groups with respect to addition.

- We prove that Q[\alpha_1] is isomorphic to Q[\alpha_2] when \alpha_1 and \alpha_2 are the two roots of the same polynomial.
- Feb 19:

We have the first mid-term exam. We will have extra office hours on Feb 18th 5 pm at the commons room. - Feb 24:

Homework 6, due in class on March 2nd.

- We go over 1st exams.
- We defined splitting field for polynomials.

- We give examples for field extensions and splitting field for f(x)\in Z_p[x].

- Feb 26:

- We introduced Eisenstein Criteria for irreducible polynomials in Q[x].
- We defined the degree of a field extension.

- Mar 2:

Homework 7 , due on March 27 th.

- We proved Eisenstein Criteria for irreducible polynomials in Q[x].
- We introduced cyclotomic extension, Q[\alpha_p] where p is a prime number.
- We went over some homework problems, especially for looking for splitting field for irreducible polynomials.

- Mar 4:

- We introduced algebraic and transcendtal element.
- We define character of a field.
- We start to discuss finite fields.

- Mar 16, 18: Extended Spring Break

- Mar 23: Notes

Homework 8 , due on April 3 rd.

- We show fields with finite elements must have finite character.
- Fields with finite elements and finite character must have p^n elements where p is the character of the finite field.
- We prove that for each prime p and integer n>0, there exits a unique finite field F with p^n elements (up to ring isomorphism). This gives a full classification for fields with finite elements.

- Mar 25: Notes

- We show that the ring homomorphism from a F_q to F_q is a group under composition. This is called the Galois group of F_q over F_p. We will get there.
- We introduced group homomorphism/group isomorphism/subgroup/coset/cyclic group/order.
- We proved Lagrange Theorem, and Fermat's Little Theorem as a consequence.

- Mar 30: Notes

- We defined normal subgroups and quotient groups(notice the remark I added at the end of notes).
- We looked at the example of S_3.

- April 1: Notes

- We proved the Fundamental Homomorphism Theorem for groups.
- We show that every permutation can be written as a product of disjoint union of cycles, and also a product of transpositions.
- We defined A_n as a subgroup of S_n.

- April 6: Notes

Homework 9 , due on April 13 rd.

- We proved that A_n contains no nontrivial normal subgroup when n is greater or equal to 5.
- We defined solvable group.
- We proved that G is solvable iff a normal subgroup N and G/N are both solvable. As a consequence, both A_n and S_n are non-solvable when n is greater or equal to 5.

- April 8: Notes

- We defined Sylow-p subgroup and introduced Sylow Theorem.
- We defined nilpotent group.
- We defined group action/transitive/stabilizer/orbit.
- We proved the stabilizer orbit formula.

- April 13: Notes

Homework 10 , due on April 22 rd.

- We proved three statements of Sylow Theorem.

- April 15: Notes

- We talk about Galois's motivation of Galois theory.
- We defined Aut(K/F)/normal extension/Galois extensions over Q/Galois group of a Galois extension.
- We proved that |Aut(F/Q)|=[F:Q] if F/Q is Galois.

- April 20: Notes

- We proved that F/Q is Galois if |Aut(F/Q)|=[F:Q].
- We proved fundamental theorem of Galois theory.

- April 22: Notes

- We proved that F/Q is Galois if and only if F is the splitting field for some polynomials in Q.
- We showed that the Galois group of a polynomial (Galois group of its splitting field) is naturally embedded into S_n where n is the degree.
- We proved (pretty hasty, the proof will not show up as part of the final exam, do not worry!) that if a polynomial is solvable with radicals, then the splitting field is solvable. This implies that when degree of f is greater or equal to 5, f generally cannot be solved with radicals!!

- April 27: Final Exam
- There will be 10 problems (open books), for 4 hours in a total (1:30 pm to 5:30 pm, one more hour than what is shown in the system for you to download/submit/etc.)
- I will be at the zoom during the exam but you don't need to be on the zoom, if there are typos in the exam, I will send emails to everyone.
- We will have the regular office hour on Thursday 11:00 am.