Examples
For example, if K is generated over ℚ by the cube root of 2, then is the product of (a copy of) K, and a splitting field of
- X3 − 2,
of degree 6 over ℚ. One can prove this by calculating the dimension of the tensor product over ℚ as 9, and observing that the splitting field does contain two (indeed three) copies of K, and is the compositum of two of them. That incidentally shows that R = {0} in this case.
An example leading to a non-zero nilpotent: let
- P(X) = Xp − T
with K the field of rational functions in the indeterminate T over the finite field with p elements. (See separable polynomial: the point here is that P is not separable). If L is the field extension K(T1/p) (the splitting field of P) then L/K is an example of a purely inseparable field extension. In the element
is nilpotent: by taking its pth power one gets 0 by using K-linearity.
Read more about this topic: Tensor Product Of Fields
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