Unhappy [Q] Baseband N9005XXUBMI6 can I root ?

I have no clue what is that Baseband, but, once I had an Xperia, was a member at xdadeveloppers and loved it...

Possible to root N9005 with my actual Baseband, yes or not ?

or ? possible to upgrade to kitkat newest version ?

sorry to be so direct at my first post, this is not very polite, but, I have so much troubles with the battery drain, my phone do not stand a full day, and, I use it very not tooo much..

I live in Hungary and 2 months ago I went to Serbia, Macedonia, Greece, Italy, Switzerland, Austria and Germany, with my old campingcar...Anyway, what I try to explain, before I left Hungary I switched to Roaming in Hungary the day before I left the contry...The next morning, my Note 3 was dead flat, I mean the battery..?? and I went to Tcom to inquire...I have no contract, but I use Tcom Domino card and I have a Sim card from them...Here, they told me I have a fake Note 3...I could not believe it and I have on it Baseband N9005XXUBMI6, Kernel 3.4.0-1628120 se.infra@R0504-11 #1, Wed Sept 11 15:24:31 KST 2013 and the Build number JSS15J.N9005XXUBMI7... Finally I crossed the border, and, this is so strange, in all the contries I went, with the Roaming, then my battery was very brave and did not drain, just normally...When I went back to Hungary, I disabled the Roaming and bingo, since about 2 weeks my phone, or my battery is draining too fast..

Did a factory restore, did many things but nothing is OK....The very last things I did yesterday I disable the MOBILE DATA and I do restore it when needed for the internet...Yes that help, but, where I do not understand, at the begining my Note 3, was working very nice....

I do not understand why so many Basebands are needed for one phone anyway..

Thanks very kindly