a) Cesarian section
b) Cesarean section
c) Caesarian section
d) Caesarean section
If you answered A, B, C or D, you are right. Well, sort of.
It turns out that variants aplenty apply to the name of this surgical procedure. Mr. Rewrite's various references books embrace some or all of the four options above.
But Mr. Rewrite refers to only one book in this case: The Associated Press Stylebook. It exists for just such a circumstances, and its role is to make the call so Mr. Rewrite and you don't have to worry about it.
AP goes with Caesarean section. That's good enough for Mr. Rewrite.
To Mr. Rewrite's frustration, however, the first AP accounts of the blessed birth in France called it a Cesarian section.
And why don't many of those paid to know better, most of whom are bound by AP style, go with it? There are hundreds of references on Google News to: Cesarian, Cesarean and Caesarian.
Why does AP use Caesarean? Mr. Rewrite doesn't care. As he's said several times now, he pays AP $25 a year for his online stylebook access so he doesn't have to worry about such things. AP isn't necessarily saying the other forms are wrong. It's picking one so its stories, assuming AP writers pay attention to the stylebook, are consistent.
You'd be wise to follow AP's lead, gentle reader.