Over an imperfect field of characteristic p, there exist irreducible inseparable polynomials

Let K be an imperfect field of characteristic p. Prove that there exist irreducible inseparable polynomials over K. Conclude that there exist inseparable finite extensions of K.


Let \alpha \in K be an element which is not a pth power in K, and let \zeta be a root of x^p-\alpha. In particular, \zeta \notin K. Now consider q(x) = (x-\zeta)^p = x^p-\alpha. Suppose t(x) is an irreducible factor of q over K. Now t cannot be linear, since \zeta \notin K. But any factor of q of degree at least 2 has a multiple root (since all the roots are \zeta). So t is an irreducible inseparable polynomial over K.

Then K(\zeta) is an inseparable extension of K.

Advertisements
Post a comment or leave a trackback: Trackback URL.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: