Page 287 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Proposition: Let F be an extension field of K, and let f(x) be a polynomial in K[x]. Then any
element of Gal(F/K) defines a permutation of the roots of f(x) that lie in F.
Let f(x) be a polynomial in K[x] with no repeated roots and let F be a splitting field for f(x) over
K. If  : K > L is a field isomorphism that maps f(x) to g(x) in L[x] and E is a splitting field for g(x)
over L, then there exist exactly [F:K] isomorphisms : F -> E such that (a) = (a) for all a in K.
Theorem: Let K be a field, let f(x) be a polynomial in K[x], and let F be a splitting field for f(x)
over K. If f(x) has no repeated roots, then |Gal(F/K)| = [F:K].

Corollary: Let K be a finite field and let F be an extension of K with [F:K] = m. Then Gal
(F/K) is a cyclic group of order m.
If we take K = Z , where p is a prime number, and F is an extension of degree m, then the
p
generator of the cyclic group Gal(F/K) is the automorphism  : F -> F defined by (x) = x , for all
p
x in F. This automorphism is called the Frobenius automorphism of F.
A symmetric function on n variables x ,..., x is a function that is unchanged by any permutation
1
n
of its variables. In most contexts, the term “symmetric function” refers to a polynomial on n
variables with this feature (more properly called a “symmetric polynomial”). Another type of
symmetric functions is symmetric rational functions, which are the rational functions that are
unchanged by permutation of variables.
The symmetric polynomials (respectively, symmetric rational functions) can be expressed as
polynomials (respectively, rational functions) in the elementary symmetric polynomials. This
is called the fundamental theorem of symmetric functions.

A function f(x) is sometimes said to be symmetric about the y-axis if f(–x) = f(x). Examples of such
functions include |x| (the absolute value) and x (the parabola).
2
31.2 Fundamental Theorem of Symmetric Functions

Any symmetric polynomial (respectively, symmetric rational function) can be expressed as a
polynomial (respectively, rational function) in the elementary symmetric polynomials on those
variables.
There is a generalization of this theorem to polynomial invariants of permutation groups G,
which states that any polynomial invariant f  R [X ,... X ] can be represented as a finite linear
1
n
combination of special G-invariant orbit polynomials with symmetric functions as coefficients,
i.e.,
f   p ( 1 ,..., n ) orbit (t),
1
G
r special
where p  R [X , ..., X ],
1
n
1
n e
1 e
and  , ...,  are elementary symmetric functions, and t = X , ..., X are special terms.
1
n
n
1
Furthermore, any special term t has a total degree  n(n – 1)/2, and a maximal variable degree
 n – 1.
31.3 Symmetric Polynomial
A symmetric polynomial on n variables x ,..., x (also called a totally symmetric polynomial) is
n
1
a function that is unchanged by any permutation of its variables. In other words, the symmetric
polynomials satisfy
f(y , y , ..., y ) = f(x , x ,..., x ), ...(1)
n
1
n
2
2
1
where y = x and  being an arbitrary permutation of the indices 1, 2, ..., n.
(i)
i
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