**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**

**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:

**Enhanced 4**^{th}
**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.