### Math 401 Spring 2021

• Office Hour: M 3:10 pm - 5:10 pm (on zoom)

• Email: wangjiuy at math dot duke dot edu

• Syllabus: Enrolled students should read carefully

• For online homework submission: GradeScope

#### Daily Update

• Jan 20:
• We introduced definition of binary operations.
• Jan 25:
Homework 1, due on Feb 1
• We introduced definition of group, group homomorphism and group isomorphism.
• We give examples of groups.
• Jan 27:
• We introduced definition of cyclic group and subgroup.
• We give examples for finite non-abelian groups.
• Feb 1:
Homework 2, due on Feb 8
• We introduced Euclidean's algorithms.
• We defined the permutation group.
• Feb 3:
• We introduced alternating group.
• We defined the normal subgroup.
• Feb 8:
Homework 3, due on Feb 15
• We introduced quotient group, kernel, image and conjugate subgroup.
• We proved fundamental theorem of homomorphism.
• Feb 10:
• We introduced direct product and solvable group.
• We proved A_n is not solvable when n>=5.
• Feb 15:
Homework 4, due on Feb 22
• We proved 2nd, 3rd group homomorphism theorem, deduce properties of solvable group, and prove S_n is not solvable when n>=5.
• We introduced group action.
• Feb 17:
• We introduced orbit, stabilizer, proved orbit stabilizer formula, and applied it to prove groups with order p^2 are all abelian.
• We stated Sylow's theorems.
• Feb 22 (no homeworks for next Monday):
• We proved Sylow's theorems.
• Feb 24:
• We introduced ring/ring homomorphism/integral domain/unit/zero-divisor.
• Mar 1:
Homework 5, due on March 8
• Midterm
• Mar 3:
• We showed that C_p is a field and proved Fermat's little theorem.
• We introduced ideals, fundamental theorem of ring homomorphisms.
• Mar 8:
Homework 6, due on March 15
• We introduced Euclidean algorithms for the ring of polynomials over a field.
• We prove that every ideal in integers and ring of polynomials over a field is generated by a single element.
• Mar 15:
Homework 7, due on March 22
• We introduced Chinese Remainder theorem and Eisenstein criteria.
• Mar 17:
• We introduced field extensions, algebraic/transcendal numbers and minimal polynomials.
• Mar 22:
Homework 8, due on March 29
• We introduced the degree and characteristic of a field.
• Mar 24:
• We introduced algebraic closure, splitting field, and started to give a classficiation for finite field.
• Mar 29:
Homework 9, due on April 5
• We introduced examples of splitting field over finite fields and over Q.
• Mar 31:
• We introduced F-automorphism of a field extension, primitive element theorem, compute Aut(E/F) for some examples.
• April 5:
Homework 10 , due on April 14
• We define Galois extensions and Galois groups, prove primitive element theorem, and show that being Galois extension is equivalent to Aut(F/Q)=[F:Q].
• April 7:
• We prove that F/Q is Galois if and only if F is a splitting field for some polynomial f(x) in Q[x], and studies examples of splitting fields.
• April 12:
• We proved the Galois correspondence theorem.
• April 14:
• We proved that quintic polynomial has no formula for roots and trisect of angle is impossible from ruler and compass. (Optional material, not required for exam)

#### Some Suggestions

• Write your homework using latex: since most of you have science majors (math, cs, economics, etc), it is likely that you will inevitably use latex in your future study. It is good to learn how to write in latex early, and keep all of your homework in a good form is rewarding.
• Discuss with your classmates: this includes asking questions to other people and answering questions for other people, you will soon find that both ways of communication benefit you.
• Do not hesitate to reach out when you start to feel you have difficulties.