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

From Groupprops

## 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 .

## Facts used

## 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] |