Let F be an extension field of K and let u F Show that Ku2

Let F be an extension field of K and let u F. Show that K(u^2 ) K(u) and [K(u) : K(uˆ2 )] = 1 or 2.

Solution

K(u²) is a subset of K(u).

Essentially, you just need to note that since a is in K(u), u² is in K(u). Any further details you should be able to work out.

Claim: [K(u) : K(u²)] = 1 or 2

Here, just note that

g(x) = x² - u²

is a polynomial with coefficients in K(u²) such that a is a root.

By the first claim,

K(u) = ( K(u²) ) (u),

Hence,

[ K(u) : K(u²) ] <= deg g = 2.