Course

Modern algebra I

You will learn groups, rings and fields: subgroups, quotients, actions, Sylow theorems, ideals and polynomial factorization. It is the basis of Galois theory, cryptography and algebraic geometry.

Recommended before this course: Discrete mathematics (1 concept comes from that course and is tagged)

47concepts

7topics

473exercises

≈ 24 hof study approx.

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Course path

1
Group theory 9 concepts
Set · Subset · Cartesian product · Function · Binary operation · Group · Identity element · Inverse element · Abelian group
2
Groups and subgroups 8 concepts
Subgroup · Order of a group · Order of an element · Generated subgroup · Cyclic group · Center of a group · Group homomorphism · Group isomorphism
3
Cosets and quotients 8 concepts
Kernel of a homomorphism · Coset · Index of a subgroup · Lagrange's theorem · Normal subgroup · Quotient group · First isomorphism theorem · Direct product of groups
4
Actions and Sylow theorems 8 concepts
Symmetric group · Cycle of a permutation · Sign of a permutation · Group action · Orbit-stabilizer theorem · Orbit and stabilizer · p-group · Sylow theorems
5
Special groups 3 concepts
Classification of finite abelian groups · Dihedral group · Simple group
6
Rings and domains 4 concepts
Unit of a ring · Zero divisor · Characteristic of a ring · First isomorphism theorem for rings
7
Polynomials and fields 7 concepts
Greatest common divisor from Discrete mathematics · Polynomial ring · Polynomial division algorithm · Euclidean domain · Irreducible element · Unique factorization domain · Field of fractions

Sources

I. N. Herstein, Topics in Algebra, 2.ª edición, John Wiley & Sons, 1975.
David S. Dummit, Richard M. Foote, Abstract Algebra, 3rd edition, Wiley, 2004.
Joseph A. Gallian, Contemporary Abstract Algebra, 9th edition, Cengage, 2017.
I. N. Herstein, Abstract Algebra, 3rd edition, Wiley, 1996.
John B. Fraleigh, A First Course in Abstract Algebra, 7th edition, Pearson, 2002.
Kenneth H. Rosen, Discrete Mathematics and Its Applications, 8th edition, McGraw-Hill, 2019.
Ralph P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction, 5th edition, Pearson, 2003.
Susanna S. Epp, Discrete Mathematics with Applications, 5th edition, Cengage, 2019.
Serge Lang, Algebra, Revised 3rd edition, Springer, 2002.