Page 279 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 3. Let P be a prime number and G be a solvable, transitive subgroup of S . Then G is a
p
subgroup of the normalizer in S of a cyclic subgroup of order ..................
p
(a) P (b) G
(c) S p (d) S
4. If f(x) be a polynomial of degree n over the field k and assume that f(x) has roots r , r ,...r n
1
2
in its splitting field F. Then element  of F defined by
(a)  = (r – r ) 2 (b)  =  (r – r) 2
2
j
3
1
1
(c)  = (r – r) j -2 (d)  =  (r – r) 3
3
j
i
1
29.3 Summary
Let f(x) be a separable polynomial over the field K, with roots r , ... , r in its splitting field
 1 n
F. Then f(x) is irreducible over K if and only if Gal(F/K) acts transitively on the roots of
f(x).
Let p be a prime number, and let G be a transitive subgroup of S . Then any non-trivial
 p
normal subgroup of G is also transitive.
Let p be a prime number, and let G be a solvable, transitive subgroup of S . Then G
 p
contains a cycle of length p.
Let p be a prime number, and let G be a solvable, transitive subgroup of S . Then G is a
 p
subgroup of the normalizer in S of a cyclic subgroup of order p.
p
Let f(x) = x + a x + · · · + a x + a be a polynomial in Q[x], and assume that
n
n-1
 n-1 1 0
a = b / d for d, b , b , ... , b in Z.
i
i
n-1
1
0
Then d f(x/d) is monic with integer coefficients, and has the same splitting field over Q as
n

f(x).
If p is a prime number, we have the natural mapping : Z[x] > Z [x] which reduces each
 p
coefficient modulo p. We will use the notation p(f(x)) = f (x).
p
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] as a
p
p
p
product of irreducible factors of degrees n , n , ... , n , then G contains a permutation with
k
2
1
the cycle decomposition
(1,2, ... ,n ) (n +1, n +2, ... , n +n ) · · · (n-n +1, ... ,n),
k
1
1
1
1
2
relative to a suitable ordering of the roots.
29.4 Keywords
Transitive Group: If G is a subgroup of the symmetric group S , then G is called a transitive
n
group if it acts transitively on the set { 1, 2, ... , n }.
Separable Polynomial: Let f(x) be a separable polynomial over the field K, with roots r , ... , r n
1
in its splitting field F. Then f(x) is irreducible over K if and only if Gal(F/K) acts transitively on
the roots of f(x).
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