In this note we give a complete proof of the following theorem of Dedekind. Our proof is a somewhat detailed version of the one given in Basic Algebra by Jacobson, Volume I,  and we shall keep the notations used in that proof.
Theorem 1 Let be square-free monic polynomial of degree n and p be a prime such that p does not divide the discriminant of . Let be the Galois group of over the field of rational numbers. Suppose that factors as:
where are distinct monic irreducible polynomials in , degree , , and .
Then there exists an automorphism which when considered as a permutation on the zeros of is a product of disjoint cycles of lengths .
2. Preliminary Results
We shall assume that the reader is familiar with the following well-known results.
1) Let be a field and be a polynomial of degree . Then any two splitting fields of are isomorphic.
2) A finitely generated Abelian group is direct sum of (finitely many) cyclic groups. (This is the fundamental theorem of finitely generated Abelian groups).
3) A system of n homogeneous equation in variables has a non-trivial solutions.
4) Let be an algebraic extension. Then any subring of containing is a subfield of . Proof: Let K be a ring such that . Let . As is algebraic over , . So .
5) (Dedekind’s Independence Theorem). Distinct characters of a monoid (a set with associative binary operation with an identity element) into a field are linearly independent. That is if are distinct characters of a monoid into a field , then the only elements , , such that
for all are , .
6) Let p be a prime and be a finite field with elements. Then the group is cyclic of order m and the generating automorphism maps to .
7) If R is a commutative ring with identity and M is a maximal ideal of R then R/M is a field.
8) Let . Then and have same cyclic structure.
Let be a polynomial of degree , and p a prime number. Then will denote the polynomial obtained by reducing the coefficients of modulo p.
Theorem 2 Let be a monic polynomial of degree and p be a prime number which does not divide the discriminant of . Let be a splitting field of over . Let be a splitting field of over . Let
where , are the roots of and
Let be the subring generated by the roots of of in . Then
• 1) There exists a homomorphism of D onto .
• 2) Any such homomorphism gives a bijection of the set R of the roots of in onto the set of the roots of the in .
• 3) If and are two such homomorphisms then there exist , such that . (Note that the restriction of to D is an automorphism of D).
Proof 1) One has that:
We claim that is a finitely generated (additive) Abelian group. Since each is a root of the monic polynomial of degree n any positive power of can be expressed as an integral linear combination of . It follows that
Therefore D is a finitely generated (additive) Abelian group generated by at most nn elements. By the Fundamental Theorem for Finitely Generated Abelian Groups D is a direct sum of finitely many cyclic groups. Since , none of these cyclic groups is finite. So D is a direct sum of finitely many infinite cyclic groups. Let be a set consisting of an independent generating system of D. We have
We claim that is a basis of . Obviously is linearly independent over . Let . Then is a ring and therefore is a field. Since for , by (4) and is a basis of . As ,
is an ideal of D and
Therefore the is finite of order . Let M be a maximal ideal of D containing pD. That is and is a finite field of characteristic p and so it has a subfield isomorphic to which we will identify as in what follows. As
the order of is , . Consider the canonical epimorphism
whose kernel is M and . Therefore . We note that as we have for
As is an epimorphism we have
is a splitting field of over . As both and are splitting fields of over they are isomorphic. Let
be such an isomorphism. Then is a homomorphism of D onto .
2) Let be a homomorphism. So . As , and has characteristic p, , so . can be extended to a homomorphism of the polynomial rings . Under this mapping . As
are the roots of the in and therefore the restriction of to R
is a bijection of the set R of roots of in to the set of the roots of in .
3) We have seen that given a homomorphism , and , is also a homomorphism from D to . We note that the restriction of to is also an automorphism of the ring D. Since , the group has order N. Let
So given a non-trivial homomorphism , we get N distinct homomorphisms , , from D to . We claim that these are all the homomorphisms from D to . Suppose that there is a homomorphism from D to which is different from , . Let us denote it by . By Dedekind Independence Theorem the set of homomorphisms from D to is linearly independent over the field .
Consider the following system of N homogeneous equations in variables ,
Since there are more variables than the equations this system of equations has a non-trivial solution. Let this non-trivial solution be , . So we have
Let . So , , . Then for we have
where . We shall show that
which will contradict the linear independence of over .
3. Proof of the Main Theorem
Since the field has order , the group has order m and where for all , is the generating automorphism of . So if is any homomorphism then so is . Since and are two homomorphisms from D to there exist such that or . This proves that the action on on is similar to the action of on . Note: In the following diagram the mapping
is the restriction of to D and we are only concerned with the effect of the mappings , and on and . Clearly
As and the effect of on is similar to the effect of on . This is further illustrated by the following: