
From: [email protected]
To: [email protected]
Date: 20050405 11:23 am
hello, sir.
i'm a student of other university.
now, i study the basic algebraic number theory.
(and i see your lecture note)
anyway, one question...
i know that some biquadratic number field has a ring of integer.
for example, ring of integer of Q( sqrt(5), sqrt(7) ) be Z( (1+sqrt(5))/2,
sqrt(7) ).
but i don't know a primitive element of ring of integer.
Z( (1+sqrt(5))/2, sqrt(7) ) =Z(alpha) , alpha = ?
help....
and i'm sorry...
i'm Korean...and i wrote in poor English.

My Response:
I think you are asking for a generator for the ring of integers of
Q(sqrt(5),sqrt(7)). Some rings of integers have a generator and some
*don't*. I don't know whether, in this particular case, yours has a
generator or not, and the obvious checks don't tell me.

CHALLENGE: Answer his question.
Solution By Dustin Clausen on April 6, 2005:
From: "Dustin Clausen"
Date: 20050406 10:02 pm
Hello Professor Stein,
I see you've got a new challenge problem up! Magma can solve this one
for you using the IndexFormEquation command. Define K to be
Q(sqrt(5),sqrt(7)) (somehow.. I couldn't figure out how to get Magma
to treat this as an absolute extension without explicitly calculating
the minimal polynomial of sqrt(5)+sqrt(7), which I did in PARI). Then
just use
O:=MaximalOrder(K);
IndexFormEquation(O,1);
and it should return an empty list, confirming that there is no a in O
such that O=Z[a]. Furthermore, if you contine testing various
IndexFormEquation(O,n), you'll see that there are a,b with [O:Z[a]]=12
and [O:Z[b]]=23, so O has no essential disciminant divisors. I wonder
if there is some other nice obstruction to rings of integers being
generated by one element? I tried to follow the link on the Magma
site to the paper that discusses the algorithm used for
IndexFormEquation, but I can't read German very well.
Regards,
Dustin

From: Claus Fieker
To: William Stein
CC: [email protected]
Date: 20050414 09:32 pm
Willam (and Dustin),
While I am unfortunately by no means an expert on index form equations I
can adress some of the problems easily:
To set the field up:
x := Polynomial([0,1]);
K1 := NumberField([x^25, x^27]);
K2 := SimpleExtension(K1);
now K2 is a absolute simple extension.
And indeed:
> IndexFormEquation(MaximalOrder(K2), 1);
[]
>
as you expected. In general however, I think that the implementation
that Klaus Wildanger did (his PhD Thesis is the German reference)
requires you to input a maximal equation order (a solution to
IndexFormEquation(?, 1);) in order to find all other solutions (up to
equivalence). For degree 4 problems the algorithm is different and is
based on simultanious ThueEquations (papers by Gaal and Pohst):
GPP93
Istvan Gaál, Attila Pethö, and Michael E. Pohst.
On the resolution of index form equations in quartic number fields.
J. Symbolic Comp., 16:563584, 1993.
GPP96
Istvan Gaál, Attila Pethö, and Michael E. Pohst.
Simultaneous representation of integers by a pair of ternary quadratic forms  With an application to index form equations in quartic number fields.
J. Number Th., 57:90104, 1996.
which are luckily in English...
Greetings
Claus
 We gave examples in class to show that the ring of integers
of a number field need not be generated by one element. Is
the ring of integers always generated by at most two elements?
Solution By Dustin Clausen on March 27, 2005:
If O_K is generated by n elements, it's a quotient of Z[X_1,...,X_n].
If 2 splits completely in K, then there are d = deg K quotients of O_K
isomorphic to F_2, each one giving rise to a distinct homomorphism
Z[X_1,...,X_n] > F_2, of which there are clearly only 2^n (one
choice for each generator).
So it all rests on finding, for any n, a number field F of degree n in
which 2 splits competely. I claim that if p is large enough and p
splits completely in Q(a) where a is a root of x^n2, then p==1 (mod
n) and we can take F to be the unique subfield of Q(\zeta_p) of degree
n. By Chebatorev's Density Theorem, there will be infinitely many
such p to choose from, so we're golden.
Proof: By "large enough" I mean it doesn't divide the index [O_{Q(a)} :
Z[a]]. In such a case we can apply Dedekind's Theorem, so p splits
completely in Q(a) if and only if x^n2 splits into linear factors (mod p).
The quotient of any two disctinct roots gives an nth root of 1 mod p, hence
there are n nth roots of 1 (mod p), hence p==1 (mod n). Also, we deduce
2^((p1)/n) == 1 (mod p) from Fermat's Little Theorem. Hence the frobenius
at 2 in F is trivial, so 2 splits completely in F, QED.
Added by William Stein:
I just looked for the first explicit example of a field K such that O_K cannot
be generated as a ring by two elements using the strategy explained above.
If we can find a number field K of degree 5 in which 2 splits completely,
then O_K cannot be generated by 2 elements. This is because, by the CRT,
O_K/(2) = F_2 + F_2 + F_2 + F_2 + F_2 (direct sum)
which is not a quotient of F_2[x,y], as there are only 4 homomorphisms
from F_2[x,y] to F_2.
So, take n=5 in Clausen's construction, and consider L=Q(2^(1/5)).
The smallest prime p that splits completely is p=151. Then Clausen
argues that if we let K be the unique degree 5 extension of Q contained in
Q(zeta_{151}), then 2 splits completely in K. Indeed, we have
2^((1511)/5) == 1 (mod 151),
so Frob_2 is trivial in K. Thus the degree 5 subfield of Q(zeta_{151}) is an
example of a field whose integer ring is not generated by 2 elements.
This is the field defined by a root of
f = x^5+x^460*x^312*x^2+784*x+128
which is the only totally real field unramified outside 151 at
http://math.la.asu.edu/~jj/numberfields/
Indeed, checking with PARI (or whatever) we see that 2 does split completely
in the above field.
William
