Very high battery drain by Android OS, Android System, and Screen in Samsung Galaxy Note 3 running Lollipop I've got a Samsung Galaxy Note 3 (SM-N900) running Android 5.0. 

The battery life is terrible. Most of the battery is taken by Android OS, Android System, and Screen. The moment I switch the screen on, the phone doesn't even last an hour. 

What should I do to fix this drain?

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This can be caused by numerous things.
Do you have any background sync going on? e.g. Google sync, few email account and so on? 


Don't sync every 5 min, try to increase the sync interval to 30 min e.g.


Do you have a bad signal reception most of the times? 


It can help to change to 2G network if thats the case.


Some other tips:
Try apps like "Greenify". It works even when you are not rooted and can terminate background processes when you don't need them.
The last thing you can do is to perform a factory reset and see if that changes anything. But you'll loose all of your data. 
An other option would be to try a second battery and see if that one performs better. 

I hope one of these advices can help you a little bit.