Page 278 - DMTH403_ABSTRACT_ALGEBRA

Unit 29: Computing Galois Groups

Let f(x) be a polynomial of degree n over the field K, and assume that f(x) has roots r , r , ... , r in Notes
2
n
1
its splitting field F. The element  of F defined by
 = (r - r) ,
2
i
j
where the product is taken over all i, j with 1  i < j  n, is called the discriminant of f(x).
It can be shown that the discriminant of any polynomial f(x) can be expressed as a polynomial in
the coefficients of f(x), with integer coefficients. This requires use of elementary symmetric
functions, and lies beyond the scope of what we have chosen to cover in the book.
We have the following properties of the discriminant:
(i)   0 if and only if f(x) has distinct roots;
(ii)  belongs to K;
(iii) If   0, then a permutation in S is even if and only if it leaves unchanged the sign of
n
 1  i < j  n(r i - r ) j 

Proposition: Let f(x) be a separable polynomial over the field K, with discriminant , and let F be
its splitting field over K. Then every permutation in Gal(F/K) is even if and only if is the square
of some element in K.
We now restrict our attention to polynomials with rational coefficients. The next lemma shows
that in computing Galois groups it is enough to consider polynomials with integer coefficients.
Then a powerful technique is to reduce the integer coefficients modulo a prime and consider the
Galois group of the reduced equation over the field GF(p).

Lemma: Let f(x) = x + a x + · · · + a x + a be a polynomial in Q[x], and assume that
n-1
n
0
1
n-1
a = b / d for d, b , b , ... , b in Z.
0
n-1
i
i
1
Then d f(x/d) is monic with integer coefficients, and has the same splitting field over Q as f(x).
n
If p is a prime number, we have the natural mapping : Z[x] > Z [x] which reduces each coefficient
p
modulo p. We will use the notation p(f(x)) = f (x).
p
Theorem [Dedekind]: Let f(x) be a monic polynomial of degree n, with integer coefficients and
Galois group G over Q, and let p be a prime such that f (x) has distinct roots. If f (x) factors in Z [x]
p
p
p
as a product of irreducible factors of degrees n , n , ... , n , then G contains a permutation with the
1
k
2
cycle decomposition
(1,2, ... ,n ) (n +1, n +2, ... , n +n ) · · · (n-n +1, ... ,n),
1
1
2
k
1
1
relative to a suitable ordering of the roots.
Self Assessment
1. IF G is a .................. of symmetric group sn then G is called transitive group. If it acts
transitively on Set {1, 2, 3, n}
(a) sub group (b) cyclic group
(c) permutation group (d) finite group
2. Let P be a prime number and let G be a transitive subgroup of Sp. Then any ..................
normal subgroup of G is also transitive.
(a) trivial (b) non-trivial
(c) finite (d) infinite
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