Different

Two quantities are said to be different (or ``unequal'') if they are not Equal.

The term ``different'' also has a technical usage related to Modules. 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

$$\Delta=\Delta(M)=\left\vert\matrix{\xi_1 & \xi_2\cr \xi_1' & \xi_2'\cr}\right\vert = \xi_1\xi_2'-\xi_1'\xi_2.$$

The different $\Delta\not=0$ Iff $\xi_1$ and $\xi_2$ are linearly independent. The Discriminant is defined as the square of the different.

See also Discriminant (Module), Equal, Module

References

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