Let
be a field and let
be the polynomial ring over
. Show that every polynomial
,
,
can be decomposed as a product
-
![{\displaystyle {}P=(X-\lambda _{1})^{\mu _{1}}\cdots (X-\lambda _{k})^{\mu _{k}}\cdot Q\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fb27f1dab00668431a5b7bc0d34960909e92fa7)
where
and
![{\displaystyle {}Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3925649ad5dc923b22e56355a5d6a466a39cb7a)
is a polynomial with no roots (no zeroes). Moreover, the different numbers
![{\displaystyle {}\lambda _{1},\ldots ,\lambda _{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f37230b2685cb565be795a02fa4243db187dec)
and the exponents
![{\displaystyle {}\mu _{1},\ldots ,\mu _{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d49f9bb391ef705cd4f5dd6ba68ebd694e26f443)
are uniquely determined apart from the order.