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Finding the subfields of a field

  1. May 2, 2017 #1
    1. The problem statement, all variables and given/known data
    Suppose that ##p(x)## is an irreducible polynomial of prime degree in ##F[x]##. What are the subfields of ##K=F[x] / \langle p(x) \rangle##?

    2. Relevant equations


    3. The attempt at a solution
    For some reason I am not seeing how to approach this problem. Some pointers would be helpful.
     
  2. jcsd
  3. May 2, 2017 #2

    fresh_42

    Staff: Mentor

    What is ##[\,K\, : \,F\,]## and what does the degree formula tells you, which you have read some minutes ago in the other thread?
     
  4. May 2, 2017 #3
    Is ##[\,K\, : \,F\,] = q##, where ##q## is the degree of ##p(x)##? Does the degree formula show that the only subfields are ##K## and the trivial subfield because ##q## is prime and can't be decomposed into factors?
     
  5. May 2, 2017 #4

    fresh_42

    Staff: Mentor

    Yes. Although I wouldn't call ##F## the trivial subfield. And you need the irreducibility of ##p(x)##, too, for otherwise the degree of the field extension would be less than ##q##.
     
  6. May 2, 2017 #5
    One more slightly related thing, that I don't really want to make another thread for. I know that ##\mathbb{Z} / \langle x^2 + 2x + 2\rangle## is isomorphic to ##\mathbb{Z} / \langle x^2 + x + 2\rangle##, because they are finite fields with order as a power of a prime, since they both have order ##3^2 = 9##. I'm just curious, is there a standard way that I could find an explicit isomorphism?
     
  7. May 2, 2017 #6

    fresh_42

    Staff: Mentor

    Let me guess: you meant ##\mathbb{Z}_3[x] / \langle x^2 + 2x + 2\rangle## and ##\mathbb{Z}_3[x] / \langle x^2 + x + 2\rangle##?
    I would determine the roots of the polynomials and map root to root. Say ##\xi^2+2\xi +2 = 0## and ##\eta^2+\eta +2 =0## then ##\varphi(\xi):=\eta## (and ##\varphi(a)=a## for ##a\in \mathbb{Z}_3##) should do.
     
  8. May 2, 2017 #7
    But aren't the polynomials irreducible?
     
  9. May 2, 2017 #8

    fresh_42

    Staff: Mentor

    Irreducibility depends on where. So, yes, they are irreducible in ##\mathbb{Z}_3[x]##, but they are not in ##\mathbb{Z}_3[\xi][x]##, resp. ##\mathbb{Z}_3[\eta][x]##. Of course, ##\xi## and ##\eta## are artificial numbers, but so is, e.g. ##\sqrt{2}## which also is only an abbreviation of a number that satisfies a polynomial equation, in this case ##x^2-2=0##.
     
  10. May 2, 2017 #9
    I see. I am little confused about the function definition though. For example, where would it send the element ##(ax+b )+ \langle x^2 + 2x +2 \rangle## for example?
     
  11. May 2, 2017 #10

    fresh_42

    Staff: Mentor

    Let's see:

    ##(ax+b )+ \langle x^2 + 2x +2 \rangle## corresponds to the element ##a\xi +b \in \mathbb{Z}_3[x]/\langle x^2+2x+2 \rangle##.
    So ##\varphi(a\xi +b) = \varphi(a)\varphi(\xi)+\varphi(b)=a\eta +b \in \mathbb{Z}_3[x]/\langle x^2+x+2 \rangle## which corresponds to the polynomial class ##(ax+b )+ \langle x^2 + x +2 \rangle##.

    To be sure, one would have to check, whether ##\varphi((ax+b) \cdot (cx+d))=\varphi(ax+b)\cdot \varphi(cx+d)## and ##\varphi((ax+b) + (cx+d))=\varphi(ax+b) + \varphi(cx+d)## hold. However, it is better to operate with ##\xi## and ##\eta##, because then it is clear, that they satisfy two different equations, whereas the difference in the multiplications of the polynomials isn't as obvious.
    So better to check ##\varphi((a\xi +b) \cdot (c\xi + d))=(a\eta +b)\cdot (c\eta +d)## and ##\varphi((a\xi +b) + (c\xi +d))=(a\eta +b) + (c\eta+d)##.

    Things become more difficult in extensions of higher degrees. But in the end, it's only a basis change of ##\mathbb{Z}_3## vector spaces. Since ##\varphi(a)=a## for all ##a \in \mathbb{Z}_3##, there is not much choice left rather than to map ##\xi \mapsto \eta## or on its conjugate, which would be the other root (that can be found by the division ##(x^2+x+2)\, : \,(x-\eta)\, = \, \ldots\;\;## Here's an example of how to do this with abstract numbers: https://www.physicsforums.com/threa...r-a-polynomial-over-z-z3.889140/#post-5595083).
     
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