Determine the splitting field and its degree over for .

Solution

We have

Since

we find that splits completely in . To see that

we note that and, as , adjoining to gives a proper extension of degree at least 2. But is a root of the quadratic , hence this extension is of degree at most 2, so it must be that its degree is precisely 2. Now is irreducible in by Eisenstein’s criterion, implying

By the multiplicative property of field extensions’ degrees, we then conclude

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