Different

In Mathematical, the different is defined in Algebraic theory of the numbers to measure the possible defect of duality of an application defined using the trace, in the ring of the entireties of a algebraic Corps of numbers $\backslash \; mathbb\; \{K\}\; \backslash ,$.

If O _{K is the ring of the entireties of $\backslash \; mathbb\; \{K\}\; \backslash ,$ and tr indicate the trace of the body of $\backslash \; mathbb\; \{K\}\; \backslash ,$ towards the body of rational numbers $\backslash \; mathbb\; \{Q\}\; \backslash ,$, then}

$tr\; (xy)\; \backslash ,$

whole quadratic form is a on O _{K . Its Discriminant as forms quadratic is not inevitably +1 (does this of it arrives only for the case $\backslash \; mathbb\; \{K\}\; =\; \backslash \; mathbb\; \{Q\}\; \backslash ,$. By defining the fractional Ideal I of $\backslash \; mathbb\; \{K\}\; \backslash ,$ like the whole of the $x\; \backslash \; in\; \backslash \; mathbb\; \{K\}\; \backslash ,$ such as tr ( xy ) is an entirety for all there in O K , then I contains O K . By definition, the different ideal $\backslash \; delta\_\; \{\backslash \; mathbb\; \{K\}\}\; \backslash ,$ is $I^\; \{-\; 1\}\; \backslash ,$, an ideal of O K .}

The standard of $\backslash \; delta\_\; \{\backslash \; mathbb\; \{K\}\}\; \backslash ,$ is the ideal of $\backslash \; mathbb\; \{Z\}$ generated by the discriminant $D\_\; \{\backslash \; mathbb\; \{K\}\}\; \backslash ,$ of $\backslash \; mathbb\; \{K\}\; \backslash ,$.

The different one can also be defined for an extension of body of numbers L / K (the different relative ) and for the local bodies. She plays a part in the Dualité of Pontryagin for the p-adic body.