Editing Determinant (section)

=== Determinants for finite-dimensional algebras===
For any [[associative algebra]] <math>A</math> that is [[dimension|finite-dimensional]] as a vector space over a field <math>F</math>, there is a determinant map
<ref>{{harvnb|Garibaldi|2004}}</ref>
:<math>\det : A \to F.</math>
This definition proceeds by establishing the characteristic polynomial independently of the determinant, and defining the determinant as the lowest order term of this polynomial. This general definition recovers the determinant for the [[matrix algebra]] <math>A = \operatorname{Mat}_{n \times n}(F)</math>, but also includes several further cases including the determinant of a [[quaternion]],
:<math>\det (a + ib+jc+kd) = a^2 + b^2 + c^2 + d^2</math>,
the [[Field norm|norm]] <math>N_{L/F} : L \to F</math> of a [[field extension]], as well as the [[Pfaffian]] of a skew-symmetric matrix and the [[reduced norm]] of a [[central simple algebra]], also arise as special cases of this construction.