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Topic: Units in algebraic integers
Replies: 18   Last Post: Sep 29, 2004 9:12 AM

 Messages: [ Previous | Next ]
 Andrzej Kolowski Posts: 353 Registered: 12/6/04
Re: Units in algebraic integers
Posted: Sep 27, 2004 9:22 AM

On 27 Sep 2004, Andrzej Kolowski wrote:
&gt;On 26 Sep 2004, James Harris wrote:
&gt;&gt;I've given the example of
&gt;&gt;
&gt;&gt;x^2 + (2s_1 + 3s_2)x + s_1 s_2 6 = (x + 2s_1))(x + 3s_2)
&gt;&gt;
&gt;&gt;where s_1 s_2 = 1, and 2s_1 + 3s_2 is an integer
&gt;&gt;
&gt;&gt;to highlight the oddity that if s_1 and s_2 are rational, then
&gt;&gt;trivially you can have both be units as both can equal 1, but if s_1
&gt;&gt;and s_2 are irrational, then they cannot be units in the ring of
&gt;&gt;algebraic integers.
&gt;&gt;
&gt;&gt;It gets weirder.
&gt;&gt;
&gt;&gt;I picked 2 and 3 because they're the first two primes--and therefore
&gt;&gt;coprime to each other--and are small, but let's replace them with 'a'
&gt;&gt;and b, non-unit coprime algebraic integers:
&gt;&gt;
&gt;&gt;
&gt;&gt;x^2 + (as_1 + bs_2)x + s_1 s_2 ab = (x + as_1))(x + bs_2)
&gt;&gt;
&gt;&gt;where s_1 s_2 = 1, as_1 + bs_2 is an integer, and ab is an integer.
&gt;&gt;
&gt;&gt;Question: Can s_1 and s_2 be irrational and algebraic integers?
&gt;&gt;
&gt;
&gt; Consider P(x) = x^2 + 55*x + 110.
&gt;
&gt; The roots are r1, r2 = (-55 +/- sqrt(3025 - 12100))/ 2.
&gt;
&gt; = 55*(-1 +/- sqrt(-3))/2.
&gt;
&gt; Letting a = 5 and b = 11, we have
&gt;
&gt; 5*s1 = -55*(-1 + sqrt(-3))/2 and
&gt;
&gt; 11*s2 = -55*(-1 - sqrt(-3))/2.
&gt;
&gt; That is,
&gt;
&gt; s1 = -11*(-1 + sqrt(-3))/2 and
&gt;
&gt; s2 = -5*(-1 - sqrt(-3))/2.
&gt;
&gt; Finally, note that (-1 +/- sqrt(-3))/2 are algebraic
&gt;integers.
&gt;
&gt; Therefore s1 and s2 are irrational and are algebraic
&gt;integers.
&gt;
&gt; But so what? Did you think this couldn't happen??
&gt;

Please ignore above. I did a wrong calculation. Here
is (I think) a valid example:

Let P(x) = x^2 + 100*x + 100.

Roots are -50 +/- 20*sqrt(6).

Let a = 2, b = 5. Then

s1 = 25 - 10*sqrt(6) and

s2 = 10 + 4*sqrt(6).

Both irrational, both algebraic integers.

Andrzej

&gt; Andrzej
&gt;
&gt;&gt;
&gt;&gt;James Harris

Date Subject Author
9/26/04 JAMES HARRIS
9/26/04 W. Dale Hall
9/26/04 magidin@math.berkeley.edu
9/26/04 W. Dale Hall
9/26/04 Bill Dubuque
9/26/04 W. Dale Hall
9/27/04 Andrzej Kolowski
9/27/04 Count Dracula
9/27/04 Dik T. Winter
9/28/04 Count Dracula
9/28/04 Dik T. Winter
9/29/04 Jesse F. Hughes
9/29/04 Dik T. Winter
9/29/04 Jesse F. Hughes
9/27/04 JAMES HARRIS
9/27/04 David Hartley
9/27/04 David Hartley
9/27/04 Andrzej Kolowski
9/27/04 Andrzej Kolowski