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email mprest@manchester.ac.uk
Spring 2010
Algebraic Structures 2 (MATH20212)
Algebraic Structures 2 : examples Just the exercises;
extracted from the notes
Euclidean algorithm for polynomials; a worked example (scanned from handwritten)
Multiplication tables of finite fields; a couple of examples
For further practice, here is (a draft version of) the August 2007 resit exam.
Past exam papers: there are 2008/9, 2007/8 and 2006/7 (papers from similarly-titled
courses in 2005/6 and earlier are not relevant). On the 2007/8 paper,
don't try to prove B12, Part 1, lines 4-5 (where it says Deduce that ...
normal subgroup of G.), because the statement is not true. (Something I had
inserted as an afterthought because the question seemed marginally light;
but it wasn't a good afterthought!)
Commutative Algebra (MATH32012)
Notes: Chapter 5 (The proofs in this chapter, apart from those of 5.2 and 5.3, are
there for completeness; in particular, they are not examinable.)
Notes: Chapter 8 (Appendix on Polynomials)
Past exam papers can be found on the school website but note that the course code and title have changed: 2007/8=MATH32012; 2006/7=MATH30512 (Polynomials); 2005/6=MATH3512. (The content also has changed a bit over the years.)
Further and associated reading:
I N Stewart and D O Tall; Algebraic Number Theory, Chapman and Hall (includes factorisation theory, noetherian rings; in some proofs substantial details are left to the reader)
Ian Stewart; Galois Theory, Chapman and Hall (not much direct overlap)
Brendan Hassett; Introduction to Algebraic Geometry, Cambridge University Press (includes varieties, Groebner Bases, Nullstellensatz)
Jean-Pierre Tignol; Galois' Theory of Algebraic Equations, World Scientific.
There are quite a few textbooks now which present Groebner basis techniques, for instance those below (the first is often recommended though note that it's directed towards algebraic geometry and goes much further that this course).
David Cox, John Little and Donal O'Shea; Ideals, Varieties and Algorithms, Springer-Verlag [Available, e.g. via the JRUL catalogue, in electronic form.]
R. Froeberg; An Introduction to Groebner Bases, John Wiley and Sons
W W Adams and P Loustaunau; An Introduction to Groebner Bases, American Mathematical Society
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