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# NUS Math Module Review: MA4292 Undergraduate topics in Mathematics 2

This course serves as an itroduction to algebraic number theory, which is concerned with studying properties of the rings of integers in a finite degree extensino of the field Q of rational numbers. The subject was developed largely in the 19th century as a result of trying to resolve Fermat’s Last Theorem. The topics which will be covered are:

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NUS Math Module Review: MA5203 Graduate Algebra I

Assessment Structure:

30% Homework (6 sets total)

20% Presentation

50% Final (Online, Open Book, No Internet)

This is the first module in the graduate algebra series (next in line are Graduate Algebra IIA/IIB), and also serves to prepare PhD students for their qualifying exam. The content mainly includes all of MA2202 Algebra I, all of MA3201 Algebra II, and about half of MA4203 Galois Theory. Prof. Zhang also included the chapters “Projective Modules” and “Hom and Duality”. The main book of reference is Algebra, Hungerford. Despite the sheer amount of content in this module, students are expected to have been exposed to concepts from both MA2202 and MA3201, and that portion of the module serve mostly as a recap.

All lectures were held via zoom (and were of course recorded). Lecture attendance was not compulsory. Like most graduate modules, this module has no tutorial classes. I only attended the first lecture, but I heard from my peers in the module that during lectures Prof. Zhang likes to ask questions to students attending lectures to make the lessons more interactive. I did not attend the remaining lectures due to my experience with him in MA3201 (refer to my review there).

(This paragraph also applies to Algebra I/II) Concepts are generally moderately difficult. My biggest issue with algebra in general is the lack of intuition and difficulty in visualising why certain results should be true (e.g. why is the quotient of a ring with a prime ideal always an integral domain?). However, this difficulty is compensated by the predictability and nice structure of algebra. That is, generally it either is not too surprising that a certain theorem holds, or provides a very nice relationship between various concepts that are not difficult to remember. Note that while this allows me to remember the results, it generally does not help when I’m trying to do the exercises. Thus, one can expect to spend a good amount of time on an algebra module in order to achieve a very good visual intuition of the results. If one is not too interested, they may choose to “black-box” the proofs of “big theorems” (e.g. fundamental theorem of Galois theory), and they should still do fine.

Homework was split into 6 parts, each of equal weightage. Homework questions are generally not straightforward, but they are all taken from the reference textbook and answers can be found online. One can expect to score very high for this component. Prof. Zhang also likes to upload some of the submitted solutions as model answers, but note that they need not be perfect (mine was used for one of the HW, despite getting 98/100).

Finals was zoom-proctored and open book/no internet, and had six questions (100 marks in total). Contrary to his exams for LA2/Algebra I/Algebra II, the questions are really difficult, but he still includes questions which require the student to state the definitions of various terms. I only managed to complete 4/6 questions with confidence.

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