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## Main Question or Discussion Point

Given [tex]K[/tex] a field, and [tex] f\in K[x][/tex] an irreducible (monic) polynomial. Does it follow that the field [tex] K[x]/\left<f\right>[/tex] is an algebraic extension of [tex]K[/tex]?

- Thread starter slamminsammya
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Given [tex]K[/tex] a field, and [tex] f\in K[x][/tex] an irreducible (monic) polynomial. Does it follow that the field [tex] K[x]/\left<f\right>[/tex] is an algebraic extension of [tex]K[/tex]?

- #2

mathwonk

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Moreover the degree (vector dimension) of the extension equals the degree of f, hence is finite, and every finite extension is definitely algebraic. so YES!

I had to think through all the details since I am old and losing my memory. hope this helps.

- #3

Hurkyl

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(Hrm. I suppose there are equivalent definitions, and some would be less obvious than others. Which are you using?)

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