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Module Discriminant

Let a module M in an integral domain D_1 for R(sqrt(D)) be expressed using a two-element basis as

M=[xi_1,xi_2],

where xi_1 and xi_2 are in D_1. Then the different of the module is defined as

Delta=Delta(M)=|xi_1 xi_2; xi_1^' xi_2^'|=xi_1xi_2^'-xi_1^'xi_2

and the discriminant is defined as the square of the different (Cohn 1980).

For imaginary quadratic fields Q(sqrt(n)) (with n<0), the discriminants are given in the following table.

-1-2^2-33-2^2·3·11-67-67
-2-2^3-34-2^3·17-69-2^2·3·23
-3-3-35-5·7-70-2^3·5·7
-5-2^2·5-37-2^2·37-71-71
-6-2^3·3-39-3·13-73-2^2·73
-7-7-41-2^2·41-74-2^3·37
-10-2^3·5-42-2^3·3·7-77-2^2·7·11
-11-11-43-43-78-2^3·3·13
-13-2^2·13-46-2^3·23-79-79
-14-2^3·7-47-47-82-2^3·41
-15-3·5-51-3·17-83-83
-17-2^2·17-53-2^2·53-85-2^2·5·17
-19-19-55-5·11-86-2^3·43
-21-2^2·3·7-57-2^2·3·19-87-3·29
-22-2^3·11-58-2^3·29-89-2^2·89
-23-23-59-59-91-7·13
-26-2^3·13-61-2^2·61-93-2^2·3·31
-29-2^2·29-62-2^3·31-94-2^3·47
-30-2^3·3·5-65-2^2·5·13-95-5·19
-31-31-66-2^3·3·11-97-2^2·97

The discriminants of real quadratic fields Q(sqrt(n)) (n>0) are given in the following table.

22^3342^3·176767·2^2
33·2^2357·2^2·5693·23
553737707·2^3·5
63·2^33819·2^37171·2^2
77·2^2393·2^2·137373
102^3·54141742^3·37
1111·2^2423·2^3·7777·11
13134343·2^2783·2^3·13
147·2^34623·2^37979·2^2
153·2^2·54747·2^2822^3·41
1717513·2^2·178383·2^2
1919·2^25353855·17
213·75511·2^2·58643·2^3
2211·2^3573·19873·2^2·13
2323·2^2582^3·298989
262^3·135959·2^2917·2^2·13
29296161933·31
303·2^3·56231·2^39447·2^3
3131·2^2655·139519·2^2·5
333·11663·2^3·119797

See also

Different, Fundamental Discriminant, Module

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References

Cohn, H. Advanced Number Theory. New York: Dover, pp. 72-73 and 261-274, 1980.

Referenced on Wolfram|Alpha

Module Discriminant

Cite this as:

Weisstein, Eric W. "Module Discriminant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ModuleDiscriminant.html

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