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I'm reading Fraleigh's A First Course in Abstract Algebra, Seventh Edition, and he makes the following definition on page 448:

A finite extension $K$ of $F$ is a finite normal extension of $F$ if $K$ is a separable splitting field over $F$.

He never explicitly defined what a normal extension is, but I've gathered from here that $K$ is a normal extension of $F$ if $K$ is a splitting field over $F$. Now, I'm trying to figure out how this relates to Fraleigh's definition of a "finite normal extension." Does his definition mean that if $K$ is a finite extension and a splitting field over $F$, then $K$ is separable over $F$? Is that even true? Please help clarify the terminology.

It seems to me like the only way to marry the various definitions would be to define a "normal extension of $F$" as a separable splitting field over $F$.

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No, a finite extension can be a splitting field, and still not be separable.

The standard example consists in taking a function field $E = F(t)$ over the field $F$ with $p$ elements, $p$ a prime. Then one takes the subfield $K = F(t^{p})$. Now $E$ is the splitting field over $K$ of the polynomial $f = x^{p} - t^{p} \in K[x]$, which can be shown to be irreducible in $K[x]$ and $E/K$ is not separable, as $f = (x - t)^{p}$ has multiple roots.

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  • $\begingroup$ Okay; so his definition of a "finite normal extension" is unrelated to the definitions of "normal extension" and "finite extension"? $\endgroup$ – justin Apr 14 '15 at 17:32
  • $\begingroup$ It seems to me like the only way to marry the various definitions would be to define a "normal extension of $F$" as a separable splitting field over $F$. $\endgroup$ – justin Apr 14 '15 at 17:42
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As Andreas Caranti pointed out, there do exist finite extensions that are splitting fields and yet are not separable. I think now that the best way to make sense of the terminology is to abandon the idea of a "finite normal extension." Leave the definition of a normal extension $E$ of $F$ as a splitting field over $F$, then based on this, define a Galois extension $E$ of $F$ as an extension that is both normal and separable. Then, we can just write "finite Galois extension" wherever Fraleigh wrote "finite normal extension."

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