# Linear representation is realizable over principal ideal domain iff it is realizable over field of fractions

## Statement

Suppose  is a Principal ideal domain (?) and  is its field of fractions. Suppose  is a linear representation of a finite group . Then, we can choose a basis for , such that, in this new basis, all the entries of the matrices  are from .

## Related facts

### Applications

In particular, this result applies to the case , and shows that for any rational representation group, we can find a representation where all the matrix entries of all the representing matrices are from .

## Proof

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Given: A linear representation  of a finite group  over the field of fractions  of a principal ideal domain .

To prove: There is a choice of basis of  in which all the matrices for  have entries from .

Proof: We let  be the vector space acted upon.

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 There exists a finite spanning set  for  (as a -vector space) such that  is -invariant  is finite-dimensional,  is finite [SHOW MORE]
2 Let  be the -submodule generated by . Then,  is a -invariant -module  is contained in . Step (1) [SHOW MORE]
3 Every element of  has a nonzero -multiple in   is the field of fractions of  (implicitly,  is an integral domain) Step (1) [SHOW MORE]
4  is a finitely generated free -module Fact (1)  is a principal ideal domain Step (2) [SHOW MORE]
5 Let  be a freely generating set for  as a -module. Then,  is a basis for   is the field of fractions of  (implicitly,  is an integral domain) Steps (3), (4) [SHOW MORE]
6 For any , the matrix for the action of  in the basis  has all its entries in  Steps (2), (5) [SHOW MORE]