Today I Learned

Some of the things I've learned every day since Oct 10, 2016

73: Eisenstein’s Criterion

Eisenstein’s criterion provide a sufficient (but not necessary) set of criterion for a polynomial with integer coefficients

P(x) = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_1 x + a_0

to be irreducible over the rationals. The criterion are that there exists a prime p such that

  1. p divides a_i where i \neq n
  2. p doesn’t divide a_n
  3. p^2 doesn’t divide a_0

If all these conditions are true, then P(x) can’t be reduced over the rationals.

Advertisements

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: