Algebraic norm in a number field - Revision history

Vipul: New page: ==Definition== The '''algebraic norm in a number field''' is a map from the number field to the field of rational numbers, defined as follows. If the number field has degree $d$
2009-01-24T01:51:29Z

New page: ==Definition== The '''algebraic norm in a number field''' is a map from the number field to the field of rational numbers, defined as follows. If the number field has degree <math>d</...

New page
==Definition==

The '''algebraic norm in a number field''' is a map from the number field to the [[field of rational numbers]], defined as follows. If the number field has degree <math>d</math>, the minimal polynomial of a given element <math>x</math> has degree <math>d_1</math>, and the constant term of its minimal monic polynomial is <math>a_0</math>, we define:

<math>N(x) = (-1)^da_0^{d/d_1}</math>.

Notice that this is not an integer-valued function on a number field; however, its restriction to the [[ring of integers in a number field|ring of integers]] is an integer-valued function, and hence a [[norm on a commutative unital ring]]. However, that norm need not necessarily be a [[nonnegative norm]].

==Facts==

* The algebraic norm in any number field is multiplicative: the norm of a product of elements equals the product of their norms. {{proofat|[[algebraic norm in a number field is multiplicative]]}}
* The norm is nonzero on all nonzero elements.