A Brief Introduction to Classical and Adelic Algebraic ... - William Stein
A Brief Introduction to Classical and Adelic Algebraic ... - William Stein
A Brief Introduction to Classical and Adelic Algebraic ... - William Stein
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176 CHAPTER 22. EXERCISES<br />
22. (*) Suppose O is an order in the ring of integers OK of a number field. Is every<br />
ideal in O necessarily generated by two elements? (Answer: No. Challenge:<br />
Given an example.)<br />
23. Find representative ideals for each element of the class group of Q( √ −23).<br />
Illustrate how <strong>to</strong> use the Minkowski bound <strong>to</strong> prove that your list of representatives<br />
is complete.<br />
24. Suppose O is an order in the ring of integers OK of a number field. Is every<br />
ideal in O necessarily generated by two elements?<br />
25. Let K be a number field of degree n > 1 with s pairs of complex conjugate<br />
embeddings. Prove that<br />
<br />
π<br />
s ·<br />
4<br />
nn<br />
> 1.<br />
n!<br />
26. Do the exercise on page 19 of Swinner<strong>to</strong>n-Dyer, which shows that the quantity<br />
Cr,s in the finiteness of class group theorem can be taken <strong>to</strong> be <br />
4 s n!<br />
π nn. 27. Let α denote a root of x3 − x + 2 <strong>and</strong> let K = Q(α). Show that OK = Z[α]<br />
<strong>and</strong> that K has class number 1 (don’t just read this off from the output of the<br />
Magma comm<strong>and</strong>s MaximalOrder <strong>and</strong> ClassNumber). [Hint: consider the<br />
square fac<strong>to</strong>rs of the discriminant of x3 −x+2 <strong>and</strong> show that 1<br />
2 (a+bα+cα2 )<br />
is an algebra integer if <strong>and</strong> only if a, b, <strong>and</strong> c are all even.]<br />
28. If S is a closed, bounded, convex, symmetric set in R n with Vol(S) ≥ m2 n ,<br />
for some positive integer m, show that S contains at least 2m nonzero points<br />
in Z n .<br />
29. Prove that any finite subgroup of the multiplicative group of a field is cyclic.<br />
30. For a given number field K, which seems more difficult for Magma <strong>to</strong> compute,<br />
the class groups or explicit genera<strong>to</strong>rs for the group of units? It is very<br />
difficult (but not impossible) <strong>to</strong> not get full credit on this problem. Play<br />
around with some examples, see what seems more difficult, <strong>and</strong> justify your<br />
response with examples. (This problem might be annoying <strong>to</strong> do using the<br />
Magma web page, since it kills your Magma job after 30 seconds. Feel free<br />
<strong>to</strong> request a binary of Magma from me, or an account on MECCAH (Mathematics<br />
Extreme Computation Cluster at Harvard).)<br />
31. (a) Prove that there is no number field K such that UK ∼ = Z/10Z.<br />
(b) Is there a number field K such that UK ∼ = Z × Z/6Z?<br />
32. Prove that the rank of UK is unbounded as K varies over all number fields.<br />
33. Let K = Q(ζ5).<br />
(a) Show that r = 0 <strong>and</strong> s = 2.