If you’re lucky enough to bump into a mathematician then you might get something along the lines of: “Algebra is the abstract encapsulation of our intuition for composition”.

These notes contain all I say in class, plus on occasion a lot more. If there are exercises in this text, you may do them but there is no credit, and you need not turn them in.

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The exercises given in this particular document are to motivate the study of abstract algebra. You should try to think about them but remember that there are no clear answers.

Read Halmos, Naive Set Theory, sections 1–15 to learn the foundations of mathematics from the point of view of set theory, and its use in formalizing the integers. Most of this should be review, although …

Jul 23, 2025 · In simple words, abstract algebra is like a toolkit for solving puzzles that involve numbers and other objects. Instead of just working with regular numbers like 1, 2, and 3, it looks at how these …

Of great historical importance is the notion of prime and the property discovered by Euclid that characterizes a prime that became a cornerstone of modern algebra.

Abstract algebra is about patterns. You see one pattern repeating itself across mathematics and you try to extract the essential elements of that pattern and turn them into a definition.

Case 1: Consider two non-parallel lines with surd coeficients; let’s call them Ax + By = C and Dx + Ey = F where A, B, C, D, E, F ∈ S. Solving these equations together (I’m omitting the algebra) gives CE − …

These are notes which provide a basic summary of each lecture for Math 331-1, the rst quarter of \MENU: Abstract Algebra", taught by the author at Northwestern University.