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The terminology would suggest that a separable field extension is so because the resulting field extension has some sort of separable topology, and that a normal extension corresponds to one with a normal topology.

I imagine this is true, or else they wouldn't have named them in such a way.

Also, I'm not sure what subfield this falls under, so if you could suggest additional tags, that would be great as well.

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Separable has a topological meaning that is not T_2. See also: mathoverflow.net/questions/7389/… –  Jonas Meyer Feb 22 '10 at 2:02
    
I see your edit has made part of my comment irrelevant, but the link is intended to get across the same point as Gerald Edgar's answer. –  Jonas Meyer Feb 22 '10 at 2:36
    
Then I misunderstood your comment. After reading it I thought there still was some connection, and mining the link for information came back with nothing, so I was still in a state of confusion. Simply saying "This is incorrect." answers my question better. –  Andrew Homan Feb 22 '10 at 2:53
    
The "normal" in "normal extension" comes from normal subgroups, or maybe the other way around. The "separable" in "separable extension" comes from separable polynomials, i.e. polynomials whose roots are distinct (hence can be "separated"). –  Qiaochu Yuan Feb 22 '10 at 3:53
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up vote 3 down vote accepted

This is incorrect. Words like "separable" and "normal" occur in unrelated ways in various parts of mathematics. (Normal subgroup, separable differential equations...) Other words, too. Like "regular", "perfect" ...

"Separable" in toplogy... Does it mean something can be "separated"? What? I believe it goes back to Fréchet, but what did he mean by it?

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So there really is no connection. Hmph. –  Andrew Homan Feb 22 '10 at 2:55
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My guess is that the connection between "separable" in topology and separation is that the fact that Q is dense in R (so R is separable) means that any two elements of R can be "separated" by an element of Q. –  Qiaochu Yuan Feb 22 '10 at 3:50
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I can "guess" things, too, but does anyone actually know? –  Gerald Edgar Mar 5 '10 at 2:46
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