Is there any connection between gezer גזר - "carrot" and the root גזר - "to cut"? I was surprised to find out there is none.
I'm not entirely sure when gezer as carrot entered Hebrew. Klein lists it as Medieval Hebrew. My guess is that it came from the Arabic jazar, which was borrowed from the Persian gazar or the Pashto gazara (from this article.) This makes sense, for the carrot originated in Afghanistan. But Persian and Pashto are Indo-European languages, not related to Hebrew.
The root גזר , meaning "to cut", on the other hand, is certainly Hebrew. We've noted before that גזר is one of a number of roots beginning with גז that mean to cut : גזה, גזז, גזר, גזל, גזם, גזע
Horowitz points out that with metathesis, גזר becomes גרז - the root of garzen גרזן - "axe".
A development from "to cut" is "to decree" - and from here we get gzar-din גזר-דין "verdict" and gezera גזרה - "decree, edict". A migzar מגזר means "a sector (of the population) and a gizra גזרה is "a shape, figure, region."
As far as the famous ancient city of Gezer - Gesenius writes that it was "probably a steep place, precipice" - separated, cut off.
In Arabic, jazira means "island" - cut off from the land. The country Algeria gets its name from here. The country is known in Arabic as Al Djazair. Stahl writes that the name comes from a number of small islands not far from the coast (which have long been connected to the land), which gave the name to the city Algiers, which gave its name to the country.
The Arabic television network Al Jazeera also means "the island" - in this case the peninsula of Qatar, surrounded by water on three sides and desert on the other.