Now, is easily seen to be Gauss's Lemma, is also irreducible over, and is thus a minimal polynomial over for } }.
32.
Let K ( \ alpha ) be an algebraic field extension generated by an element \ alpha, which has P ( x ) as minimal polynomial.
33.
If is a matrix with positive eigenvalues and minimal polynomial, then the Jordan decomposition into generalized eigenspaces of can be deduced from the partial fraction expansion of.
34.
This implies in particular that an integral element over an integrally closed domain " A " has a minimal polynomial over " A ".
35.
In linear algebra, the "'minimal polynomial "'of an field is the monic polynomial over of least degree such that 0 } }.
36.
The minimal polynomial always divides the characteristic polynomial, which is one way of formulating the Cayley Hamilton theorem ( for the case of matrices over a field ).
37.
A " separable extension " is an extension that may be generated by " separable elements ", that is elements whose minimal polynomials are separable.
38.
By taking appropriate sums of conjugates of, following the construction of Gaussian periods, one can find an element of that generates over, and compute its minimal polynomial.
39.
By finding the minimal polynomial of x in this ring, we have computed p _ i ( x ), and thus factored p ( x ) over K.
40.
The minimal polynomial of " x " divides every polynomial which has " x " as a root ( this is Abel's irreducibility theorem ).
