ABOUT K.ring_of_integers()

number_theory

asked 2013-12-05 03:13:24 +0200

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in William Stein book PAGE 31Algebraic Number Theory,a Computational Approach

there can run out the module basis derectly,but I try in sagenb online,there no module basis,why?

sage: K.<a> = QuadraticField(5)
sage: OK = K.ring_of_integers(); OK
Order with ***module basis 1/2*a + 1/2***, a in Number Field
in a with defining polynomial x^2 - 5
sage: Frac(OK)
Number Field in a with defining polynomial x^2 - 5
----------------------------------

K.<a> = QuadraticField(5);K;OK = K.ring_of_integers(); OK

Number Field in a with defining polynomial x^2 - 5
Maximal Order in Number Field in a with defining polynomial x^2 - 5

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answered 2013-12-15 01:36:07 +0200

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thank you very much!

I was mistake the big blue number is a basis of QuadraticField(10000019); I have known the second question,big number only is a example

I try use CALC

frattini(10000019,44017415641172741107677902381027222931990967264900) even period length 442 X1=51728669288581496747017067236834679830362903437357520297507560505871495808089 39912744279034480986438365128783512278562690868566790783049793210477650310733452 599026227120591649690086336036036403311756634562204182936222240930,Y1=1635802598 80346328225592238121094625499142677693142915506747253000340064100365767872890438 81624927126642399817503030943657561063163927237760168060379588379147781761197418 40754457028237899759459100428895693238165048098039 10852834095525680664513927699041160260579360110501132510090865173423602690330249 77812047055639758593737342159850352766144823554909022626357346610325892328298159 61678064598952331111420248058645481992204150696013354372444863354013306453069185 36270.00

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answered 2013-12-13 12:35:10 +0200

John Cremona
701 ●35 ●22

Your original question was (I think) about how to get a module basis for an order. The output format seems to have changed since the book from which you quote was written, but it is still possible:

sage: K.<a> = QuadraticField(5)
sage: OK = K.ring_of_integers(); OK
Maximal Order in Number Field in a with defining polynomial x^2 - 5
sage: OK.basis()
[1/2*a + 1/2, a]

The last line gives a Z-basis for OK.

I don't know what you are asking in the follow-up question. Can you explain?

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answered 2013-12-05 04:18:37 +0200

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page 94,I run no the result....

sage: K.163580259880346328225592238121094625499142677693142915506747253000 340064100365767872890438816249271266423998175030309436575610631639 272377601680603795883791477817611974184075445702823789975945910042 8895693238165048098039 a - 517286692885814967470170672368346798303629034373575202975075605058 714958080893991274427903448098643836512878351227856269086856679078 304979321047765031073345259902622712059164969008633603603640331175 6634562204182936222240930*

G = K.unit_group ();G;G.0;G.1;

Unit group with structure C2 x Z of Number Field in a with defining polynomial x^2 - 10000019 u0 u1

(G.1).list()

[0, 1]

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ABOUT K.ring_of_integers()

there can run out the module basis derectly,but I try in sagenb online,there no module basis,why?

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there can run out the module basis derectly,but I try in sagenb online,there no module basis,why?

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ABOUT K.ring_of_integers()