Mathematics 571
Number Theory
Spring 2006
Problem Set 1. 1. Find a monic polynomial with integer coecients which has 5+ 7 as a root.
2. Show that given integers r1 0, r2 0, r1 + r2 1 there exists a number eld K such that K Q R = Rr1 Cr2 . 3. Let D be a sq

Mathematics 571
Number Theory
Spring 2006
Problem Set 8. 1. (Chinese Remainder Theorem) Let R be a commutative ring with identity and let A1 , . . . , An be ideals of R such that Ai + Aj = R when i = j . Show that given x1 , . . . , xn in R, there exists

Mathematics 571
Number Theory
Spring 2006
Problem Set 7. 1. Let K be a number eld of degree n. a) Suppose that p < n is a rational prime which splits completely as a product of n distinct prime ideals in OK . Show that for any OK such that K = Q(), the in

Mathematics 571
Number Theory
Spring 2006
Problem Set 6. 1. Let K be a number eld. An element of OK is called irreducible if whenever = in OK , then at least one of , is a unit of OK . a) Show that is irreducible if and only if the factorization of the id

Mathematics 571
Number Theory
Spring 2006
Problem Set 5. 1. Show that the ideal generated by the norm N (A) of an ideal A in the maximal order of a number eld is divisible by A. 2. Show that the maximal order in Q( 6) is a principal ideal domain. 3. Consi

Mathematics 571
Number Theory
Spring 2006
Problem Set 4. 1. Use the Minkowski lattice point theorem to prove that every prime congruent to 1 modulo 6 can be written in the form x2 +3y 2 . Show similarly that all primes congruent to 1 modulo 8 are of the f

Mathematics 571
Number Theory
Spring 2006
Problem Set 3. 1. (Stickelberger) Show that the discriminant of an order in a number eld has remainder 0 or 1 after division by 4. Hint: Use the denition of the discriminant as the square of a determinant , and us

Mathematics 571
Number Theory
Spring 2006
Problem Set 2. Recall that the eld Qp is the completion of the rational numbers with respect to the p-adic metric |p . 1. Show that the subset Zp Qp consisting of all elements z Qp such that |z |p 1 is the maximal

Mathematics 571
Number Theory
Spring 2006
Problem Set 12. 1. Let K be a number eld with maximal order OK . Show that there are innitely many prime ideals P OK such that N P is prime (that is, the residue degree fP = 1). Hint: Consider the behavior of K (s

Mathematics 571
Number Theory
Spring 2006
Problem Set 11. 1. Let l and p be odd primes with l 1 (mod 3). a) Show that l splits completely in Q( 3 p) if and only if p splits completely in the cubic subeld of the cyclotomic eld Q(l ). b) Show that if 2 is a

Mathematics 571
Number Theory
Spring 2006
Problem Set 10. 1. Find all integers x, y which solve the equation 3x2 4y 2 = 11 by introducing a suitable order O in a number eld and analyzing it. 2. Let K be a number eld.We say that K is a CM eld if K is total

Mathematics 571
Number Theory
Spring 2006
Problem Set 9. Remark: For K a number eld, we abuse terminology by referring to the units in OK as units of K . By fundamental units of K we of course mean a basis for O /(O )tor K K as a free Abelian group. 1. Su