### Math 401 Spring 2020

• 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

#### Daily Update

• 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.
• 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.
• Jan 15:
• We prove that Z_m is a ring.
• We introduced the definitions of zero-divisor, unit, integral domain and field.
• 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.
• Jan 29:
• We gave a comparison between Z and F[x].
• We introduced subring, kernel, ideal, principal ideals.
• 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.
• Feb 5:
• We discussed Z_m as quotient rings of integers.
• We proved fundamental homomorphism theorems.
• 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.
• 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.
• 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.
• 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.
Good luck! Let me know if you have other questions about the arrangement of exams.