# Reduced norm

From Encyclopedia of Mathematics

Let be a finite-dimensional central simple algebra over . A finite extension field of is a splitting field for if as -algebras for some . Here is the -algebra of -matrices. Choose an isomorphism . The reduced norm mapping is now defined by

and the reduced trace mapping is similarly defined by

One checks that the right-hand sides of these equations are indeed in (and not just in ) and that the definitions are independent of the choices of and .

The reduced norm is multiplicative, and is invertible if and only if . The reduced trace is a homomorphism of vector spaces, and defines a non-degenerate bilinear form on .

#### References

[a1] | H. Bass, "Algebraic -theory" , Benjamin (1967) pp. 152ff |

[a2] | A.J. Hahn, O.T. O'Meara, "The classical groups and -theory" , Springer (1979) pp. §2.2D |

**How to Cite This Entry:**

Reduced norm.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Reduced_norm&oldid=13357