Do Irreducibles Induce Algebraic Extensions?

  1. 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. jcsd
  3. mathwonk

    mathwonk 9,956
    Science Advisor
    Homework Helper

    i think so. in general, if M is a maximal ideal in a commutative ring R, then R/M is a field. If R contains a subfield k, then R/M is an extension of k. So in your case the ring K[X] contains the subfield K, and since K[X] is a Euclidean ring, hence also a p.i.d., an irreducible polynomial f generates a maximal ideal, so K[X]/(f) is a field extension of K.

    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.
  4. Hurkyl

    Hurkyl 15,987
    Staff Emeritus
    Science Advisor
    Gold Member

    Isn't the answer obvious from the definition of "algebraic extension"? The interesting part is that it's a field and not just a ring.

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