Let
a be an element of some
field F and let
k be a
subfield of
F. Suppose that
a is a root of some nonzero
polynomial in
k[x]. The minimal
polynomial of
a over
k
is the monic (i.e. x
n+lower degree terms)
polynomial of least degree in
k[x] that has
a as a
root.
Here are some properties of the minimal polynomial
-
It is unique. (For suppose that f and h are both minimal
polynomials of a over k. Then f-h has lower degree
than f and h and has a as a root. If it is not zero
this contradicts the definition of minimal polynomial.)
-
It is irreducible. (If not then one of its factors has smaller degree
and has a as a root, again contradicting the definition.)
-
If h(x) is a nonzero polynomial over k that has a
as a root then the minimal polynomial is a factor of h(x). (Similarly.)
Examples
The Cayley-Hamilton Theorem shows that an nxn matrix is a zero
of a polynomial. It follows that there is an analogous notion of minimal
polynomial for matrices. The derivation of the Jordan canonical form
for matrices uses the minimal polynomial.