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Algebraic number theory
Final Exam
Sujatha
Answer the following:
(3 × 5 = 15)
(You are allowed to quote and use theorems, but otherwise full details
should be provided).
√
√
1) Let K = Q( d). Show that if 4|d + 1, then (−1±2 −d) is an algebraic
integer.
√
2) What is the ring of integers OK in K = Q( −5) ?. Factorise the ideal
(6) in OK into prime ideals.
3) K is a finite extension of Q. Show that for m ∈ Z and a ∈ OK , d(a+m) =
d(a).1
4) Let F ⊂ K ⊂ L be a sequence of number fields, and let A ⊂ B ⊂ C be
their rings of integers. If P ⊂ Q ⊂ R are prime ideals in A, B and C
respectively, show that
R
R
Q
R
R Q
e( )e( ) = e( ); f ( )f ( ) = f ( )2
Q P
P
Q
P
P
5) Let a be an element of a number field K. Define the norm of a. Prove
or disprove: N (a) = ±1 =⇒ a is a unit in OK .
(4 × 15 = 60)
Answer any four of the following
1. (a) Define a dedekind domain and give two examples. Is Z[x, y] a
dedekind domain ? Justify your answer.
(4)
(b) Let A ⊃ B 6= 0 be two ideals in a dedekind domain. Show that
A = B + (a) for some a ∈ R.
(3)
1
d may mean degree/discriminant with respect to 1, a, . . . an−1 - not clear
Q
e
e( Q
P ) is the largest integer e such that Q ⊃ P and f ( P ) is the degree of the extension
A
over P
2
B
Q
1
(c) Define a fractional ideal in a dedekind domain. For any nonzero ideal A in a dedekind domain, show that there exists a nonzero ideal B such that AB is principal.
(4)
(d) Show that any integral domain containing a field k and algebraic over k is itself a field.
(4)
2.
• Define the class group of a number field K. Show that x2 + y 2 =
p has a solution in Z iff p ∼
(5)
= 1 mod (4).
• Let OK be the ring of integers of a number field. Show that each
equivalence class of ideals has an integral ideal representative.
(5)
• Assuming the finiteness of the class group, show that for any
algebraic number field K, there is a finite extension L of K such
that every ideal of OK becomes principal in OL .
(5)
3.
• Let K be an algebraic number field. State the theorem on Minkowski’s
bound
the bound to show that
√ for the norm of ideals in OK . Use√
Q( −5) has class number 2 and that Q( 5) has class number 1.
(8)
• Let K = Q(a) where a is a root of x3 + x + 1. Exhibit an integral
basis for K and compute the discriminant of K. What are the
primes that ramify in K ?
(7)
4.
• Define the different D of an algebraic number field K and show
that it is an ideal in OK while its inverse is a fractional ideal.
What is the norm of D ?
(8)
• State Dirichlet’s unit theorem. Show that upto multiplication by
units, there are only finitely many elements α ∈ OK of a given
norm a ∈ Z.
(4)
5.
• Show that the group of roots of unity in K is cyclic of even order.
(3)
• State the law of quadratic reciprocity. Describe p2 . Let q be
an odd prime congruent to 1 mod 4. Show that q is a quadratic
residue mod p iff p ∼
= r mod q where r is a quadratic residue
mod 8.
(7)
• Show that there are infinitely many primes of the form 8k + 7.
(4)
2
• Show that an integer n is properly represented by some binary
form of discriminant d iff the congruence u2 ∼
= d mod 4n is solvable for some u ∈ Z. Give two examples of imaginary quadratic
fields with class number 1.
(4)
3