Feauture selection for clustering regarding zero-correlated feature
Lack of correlation with other features is not a reason to omit a feature. On the contrary, it is usually a reason to keep the feature because it may provide unique information. Typically, highly correlated features provide redundant information and feature reduction techniques (e.g., Principal Components Analysis) are used to remove the redundancy.

While it is possible that the uncorrelated feature is noise, you should not make that assumption. It could be that the uncorrelated feature is the only one containing information and the other 4 features are all correlated noise.