Algebraic varieties | K-theory
In mathematics, a norm variety is a particular type of algebraic variety V over a field F, introduced for the purposes of algebraic K-theory by Voevodsky. The idea is to relate Milnor K-theory of F to geometric objects V, having function fields F(V) that 'split' given 'symbols' (elements of Milnor K-groups). The formulation is that p is a given prime number, different from the characteristic of F, and a symbol is the class mod p of an element of the n-th Milnor K-group. A field extension is said to split the symbol, if its image in the K-group for that field is 0. The conditions on a norm variety V are that V is irreducible and a non-singular complete variety. Further it should have dimension d equal to The key condition is in terms of the d-th Newton polynomial sd, evaluated on the (algebraic) total Chern class of the tangent bundle of V. This number should not be divisible by p2, it being known it is divisible by p. (Wikipedia).
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