Algebraic number theory M3P15/M4P15
sheet 1 sheet 2 sheet 3 sheet 4 sheet 5
(The problem sheets are the same as last year. Solutions will appear here in due course.)
solutions 1 solutions 2 solutions 3 solutions 4 solutions 5
test 1 solutions to test 1
test 2 solutions to test 2
Office hour: Fridays 12 pm to 1 pm in room 664
I can recommend this short and inexpensive book: Pierre Samuel, Algebraic Theory of Numbers
(Be warned that it has more material than lectures and is rather tersely written.)
Some old lecture notes on basic algebra:
lectures on rings and fields (you can ignore Chapters 6 and 7)
Tests and solutions from last year:
test 1 solutions for test 1
test 2 solutions for test 2
Enhanced 4th year course work: write on Cyclotomic fields.
Deadline for submission: Friday 4 May
Calculate the ring of integers, the discriminant, the group of units;
classify prime ideals, explaining how the principal ideal generated by a prime
number is written as the product of prime ideals. If you still have time and energy,
you can then explore the relation to the Fermat’s last theorem, or the so called
“cyclotomic units”, or the class group of a cyclotomic field (the Vandiver conjecture).
Try to be original and write something interesting, but make sure that your mathematics
is precise and rigorous.