Page 168 - Advanced Linear Algebra

152 Advanced Linear Algebra

Since  ? , we have ~ and there exist  9 and !  : for which

" ~ # b !
Hence,

" ~ " ~² # b !³ ~ # b !

Now we need more information about . Multiplying the expression for " by

c gives

~ " ~ c ² "³ ~ c # b c !

and since ºº#»» q : ~ ¸ ¹ , it follows that c # ~ . Hence, “ c , that

is, “ and so ~ for some  9 . Now we can write

" ~ # b !
and so

²" c #³ ~ !  :
Thus, we take ~ to get (6.2) and that completes the proof of existence.

For uniqueness, note first that 4 has orders and and so and are

)
associates and ~ . Next we show that ~ . According to part 2 of
Lemma 6.10,
4 ² ³ ~ ºº# »» ² ³ l Ä l ºº# »» ² ³

and
4 ² ³ ~ ºº" »» ² ³ l Ä l ºº" »» ² ³

where all summands are nonzero. Since 4 ² ³ ~ ¸ ¹ , it follows from Lemma
6.10 that 4 ² ³ is a vector space over 9 ° º » and so each of the preceding
decompositions expresses 4 ² ³ as a direct sum of one-dimensional vector
subspaces. Hence, ~ dim ²4 ² ³ ³~ .
Finally, we show that the exponents and are equal using induction on . If

~ , then ~ for all and since ~ , we also have ~ for all .

Suppose the result is true whenever  c and let ~ . Write

² ÁÃÁ ³ ~ ² ÁÃÁ Á ÁÃÁ ³Á €

and
² ÁÃÁ ³ ~ ² ÁÃÁ Á ÁÃÁ ³Á €

!
!