Are "polite" and "politics" etymologically similar?

I read of the words "polite" and "politics" on Wiktionary. They originate from Latin word for "smooth" and Greek word for "state", so superficially i concluded they have nothing in common.

But the connection makes sense. Wiktionary defines "politics" also as "Political maneuvers or diplomacy between people, groups, or organizations, especially involving power, influence or conflict". To be successful with politics an individual has to be polite, woo the crowd, be a nice guy. Rude and hostile people in such environment are quickly marginalized and shut up.

Were "polite" and "politics" connected etymologically at some point?

They are not related.

Polite (en) <- Latin politus "polished" <- Latin polio "polish" <- *pel- "to strike" (Proto-Indo-European) (see Wiktionary)

Politics (en) <- Politic (en) <- Polis "city" (ancient Greek) <- *tpolH- "fortification" (Proto-Indo-European) (see Wiktionary)

I always thought that Polite(en) comes from Polites(gr) which means citizen.

Being Polite in ancient greece, meant to be a citizen and behave like one.This might have included voting, having rights,having mannerisms, participating in social affairs , etc.

Accordingly, when being told you are not Polite was a derrogatory term. It meant you do not behave like a citizen and you are standing very far from the ideals of ones society.

political life in Greece was of high importance and anyone who did not participate was being called an Idiotis(gr) or Idiot (en) which meant "privateer" or one who only minds his own business and doesnt bother with society.