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Unformatted text preview: 444 9. FIELD EXTENSIONS – SECOND LOOK Deﬁne a total order on ntuples of nonnegative integers and, in particular, on partitions by ˛ > ˇ if the ﬁrst nonzero difference ˛i ˇi is positive.
Note that
. / for any permutation . This total order on ntuples ˛
induces a total order on monomials x ˛ called lexicographic order.
Lemma 9.6.7.
(a) The leading (i.e., lexicographically highest) monomial of
x.
(b)
X
.x1 ; : : : ; xn / D
T m .x1 ; : : : ; xn /; is j jDj j where T is a nonnegative integer, T
>. D 1, and T D 0 if (c)
X mD S .x1 ; : : : ; xn /; j jDj j where S is a nonnegative integer, S
>. D 1, and S D 0 if Proof.
.x1 ; : : : ; xn / D .x1 x 1 /.x1 D x1 1 x2 2 xn n C x 2/ .x1 x n/ C : : : ; where the omitted monomials are lexicographically lower, and j is the
number of k such that k
j ; but then j D j , and, therefore, the
leading monomial of
is x . Since
is symmetric, we have
D
m C a sum of m , where j j D j j and <
in lexicographic order.
This proves parts (a) and (b).
Moreover, a triangular integer matrix with 1’s on the diagonal has an
inverse of the same sort. Therefore, (b) implies (c).
I
Example 9.6.8. Take n D 4 and
We have
D . 4 /2 . 3 /2 D .4; 4; 3; 3; 2/. Then D .5; 4; 4; 2/. 1 D .x1 x2 x3 x4 /2 .x1 x2 x3 C x1 x3 x4 C x2 x3 x4 /2 .x1 C : : : x4 /
5442
D x1 x2 x3 x4 C : : : ;
5442
where the remaining monomials are less than x1 x2 x3 x4 in lexicographic
order. ...
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 Fall '08
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 Algebra, Integers

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