Page 475 - Advanced Linear Algebra
P. 475
An Introduction to Algebras 459
monic polynomial ²%³ of smallest degree that is satisfied by is called the
minimal polynomial of .
-
If ( is algebraic over , then the subalgebra generated by over is
-
-´ µ ~ ¸ ² ³ ²%³ -´%µÁ deg ² ³ deg ² ³¹
and this is isomorphic to the quotient algebra
-´%µ
-´ µ
º ²%³»
where º ²%³» is the ideal generated by the minimal polynomial of . We leave
the details of this as an exercise.
The minimal polynomial can be used to tell when an element is invertible.
Theorem 18.4
)
1 The minimal polynomial ²%³ of ( generates the annihilator of ,
that is, the ideal
ann² ³~¸ ²%³-´%µ ² ³~ ¹
of all polynomials that annihilate .
)
2 The element ( is invertible if and only if ²%³ has nonzero constant
term.
Proof. We prove only the second statement. If is invertible but
²%³ ~ % ²%³
then ~ ² ³ ~ ² ³ . Multiplying by c gives ² ³ ~ , which contradicts
the minimality of deg² ²%³³ . Conversely, if
²%³ ~ b % b Ä b c % c b %
where £ , then
~ b bÄb c c b
and so
c c c
² b bÄb c b ³ ~
and so
c
c ~ ² b b Ä b c c b c ³
