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
Famous quotes containing the word examples:
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold peoples attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (18961966)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (16701733)