1

Are there cases in which a polynomial cannot be written in a polynomial split? So can any P(x) be written in the form $P(x) = (-1)^n(x-a_1)^{k_1}...(x-a_p)^{k_p}$?

Ken Duna
  • 5,746
aribaldi
  • 1,400
  • Could you explain further. Are you talking about splitting fields? Your tag is linear-algebra. – almagest May 20 '16 at 14:24
  • What do you mean by "in a polynomial spilt"? Are you asking for polynomials that cannot be written as a product of simpler polynomials? That would be the case for $x+1$, for example. Whether there are examples of degree $>1$ depends on which kind of numbers you accept as coefficients. – hmakholm left over Monica May 20 '16 at 14:25

1 Answers1

2

Yes. There are two factors which affect whether or not a polynomial $P(x)$ can be split or not:

  1. The polynomial $P(x)$ itself.
  2. The field $F$ that the polynomial is over.

For example, the polynomial $P(x) = x^2 + 1$ does not split over $\mathbb{R}$, but it does split into $P(x) = (x - i)(x + i)$ over $\mathbb{C}$.

Furthermore, if $P(x)$ is a polynomial with entries in a field $F$ and $F$ is an algebraically closed field then $P(x)$ splits over $F$.

If $F$ is not algebraically closed, then there exists a field extension $K$ > $F$ such that $P(x)$ does split over $K$.

Irregular User
  • 3,862