Page 288 - DMTH403_ABSTRACT_ALGEBRA

Unit 31: The Galois Group of a Polynomial

For fixed n, the set of all symmetric polynomials in n variables forms an algebra of Notes
dimension n. The coefficients of a univariate polynomial f(x) of degree n are algebraically
independent symmetric polynomials in the roots of f, and thus form a basis for the set of all such
symmetric polynomials.
There are four common homogeneous bases for the symmetric polynomials, each of which is
indexed by a partition  (Dumitriu et al., 2004). Letting l be the length of , the elementary
functions e , complete homogeneous functions h , and power-sum functions p are defined for



l = 1 by
e =  x ...x ...(2)
 1 1 j j  1
2 j
1 j    1
... j 
n
h =   x m j ...(3)
 1 m    
1 ... mn l1 j 1
n
p =  x . ...(4)


 1 j 1

and for l > 1 by
l
s = s i  ...(5)
 
i 1

where s is one of e, h or p. In addition, the monomial functions m are defined as
m = x  1  (1) x  s (2) ... x   m (m ) , ...(6)
 
2
 S 
where S is the set of permutations giving distinct terms in the sum and  is considered to be

infinite.
As several different abbreviations and conventions are in common use, care must be taken when
determining which symmetric polynomial is in use.
The elementary symmetric polynomials  (x , ..., x ) (sometimes denoted  or e ) on n variables
k
1
n
k

{x , ..., x } are defined by
n
1
 (x , ..., x ) =  x i ...(7)
1
1
n
1 i n


 (x , ..., x ) =  x x j ...(8)
n
i
1
2
1 i j n



 (x , ..., x ) =  x x x k ...(9)
i
1
3
n
j
1 i j k n




 (x , ..., x ) =  x x x x l ...(10)
k
i
j
4
n
1
1 i j k l n





 
 (x , ..., x ) =  x i ...(12)
n
1
5
1 i n


The kth elementary symmetric polynomial is implemented in Mathematica as Symmetric
Polynomial [k, {x , ..., x }]. Symmetric Reduction [f, {x , ..., x }] gives a pair of polynomials
n
1
1
n
{p, q} in x , ..., x where is the symmetric part and is the remainder.
1
n
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