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Let $A \subseteq B \subseteq C$ be fields and let $\alpha$, $\beta$, $\gamma$ be such that $A(\alpha) = B$, $B(\beta) = C$, $A(\gamma) = C$. Assume $B$ and $C$ have finite degree over $A$.

Let $m(\alpha,A)$ be the minimal polynomial of $\alpha$ over $A$, let $m(\beta,B)$ be the minimal polynomial of $\beta$ over $B$, let $m(\beta,A)$ be the minimal polynomial of $\beta$ over $A$, and let $m(\gamma,A)$ be the minimal polynomial of $\gamma$ over $A$.

Let $c \in C$. We have, on the one hand, $$ c=\sum_{i = 1}^{[C:B]} b_i \beta^{i-1}, \quad b_i \in B $$ $$ b_i=\sum_{j = 1}^{[B:A]} a_{ij} \alpha^{j-1}, \quad a_{ij} \in A $$ and on the other hand $$ c=\sum_{k = 1}^{[C:A]} c_k \gamma^{k-1}, \quad c_k \in A $$ We also have $$ \gamma = \sum_{i = 1}^{[C:B]} d_i \beta^{i-1}, \quad d_i \in B $$ $$ d_i=\sum_{j = 1}^{[B:A]} e_{ij} \alpha^{j-1}, \quad e_{ij} \in A $$ If the polynomials $m(\alpha,A)$, $m(\beta,B)$, $m(\beta,A)$, $m(\gamma,A)$ and the numbers $a_{ij}$, $e_{ij}$ are known explicitly, how can I calculate the $c_i$?

ADDED: Now that Joriki and Jyriki have helped me to formulate the problem correctly, I see the solution is not so hard. Since $$ \gamma = \sum_{i = 1}^{[C:B]} \sum_{j = 1}^{[B:A]} e_{ij} \alpha^{j-1}\beta^{i-1} $$ we can find numbers $f_{ijk}$ such that $$ \gamma^{k-1} = \sum_{i = 1}^{[C:B]} \sum_{j = 1}^{[B:A]} f_{ijk} \alpha^{j-1}\beta^{i-1} $$ by using the polynomials $m(\alpha,A)$, $m(\beta,B)$ to express large powers of $\alpha$ and $\beta$ in terms of smaller powers. Then $$ c = \sum_{i = 1}^{[C:B]} \sum_{j = 1}^{[B:A]} a_{ij} \alpha^{j-1}\beta^{i-1} $$ and also $$ c = \sum_{k=1}^{[C:A]} c_k \left( \sum_{i = 1}^{[C:B]} \sum_{j = 1}^{[B:A]} f_{ijk} \alpha^{j-1}\beta^{i-1} \right) = \sum_{i = 1}^{[C:B]} \sum_{j = 1}^{[B:A]} \left( \sum_{k=1}^{[C:A]} c_k f_{ijk} \right) \alpha^{j-1}\beta^{i-1} $$ Therefore $$ a_{ij} = \sum_{k=1}^{[C:A]} c_k f_{ijk} $$ So we are reduced to solving this system of system $[C:A] = [C:B] \cdot [B:A]$ linear equations for the $c_k$.

maxpower
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  • With some difficulty, I think. Take a fairly simple example. Let $A$ be the rationals, $\alpha=\sqrt2$, $\beta=\sqrt3$, $\gamma=\sqrt2+\sqrt3$. You have $c=a_{11}+a_{12}\sqrt2+a_{21}\sqrt3+a_{22}\sqrt6$, and you want $c=c_1+c_2(\sqrt2+\sqrt3)+c_3(\sqrt2+\sqrt3)^2+c_4(\sqrt2+\sqrt3)^3$. Expressing the $c_i$ in terms of the $a_{ij}$ (and the minimal polynomials) looks mildly unpleasant. – Gerry Myerson Sep 08 '11 at 06:30
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    I would have thought that you can't, since there may be several values of $\gamma$ with the same $A(\gamma)$ and $m(\gamma,A)$ that will generally have different $c_i$? For example, consider $A=\mathbb Q$, $\alpha=1$, $\beta=\sqrt2$ and $\gamma=\pm\sqrt2$. Then $A(\gamma)=\mathbb Q(\sqrt2)$ and $m(\gamma,A)=x^2-2$ for both signs, but the sign of $c_2$ is flipped. – joriki Sep 08 '11 at 06:46
  • @Gerry Unpleasant or not, I need to find an algorithm do it. – maxpower Sep 08 '11 at 09:20
  • @joriki I don't think this shows it can't be done. You know the $a_{ij}$ after all. – maxpower Sep 08 '11 at 09:22
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    @maxpower: I don't understand. How do the $a_{ij}$ help? They're the same for both signs in my example. – joriki Sep 08 '11 at 09:29
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    @maxpower: I agree with joriki. The problem specification must include a way of identifying the element $\gamma$ in terms of $\alpha$ and $\beta$. Another example would be $\alpha=\sqrt2$, $\beta=i$, $\gamma$ any primitive eighth root of unity. All four sign combination are possible in $\gamma=\pm\alpha(1\pm\beta)/2$ in that all those numbers share the same minimial polynomial $x^4+1$. Unless you know which combination is $\gamma$, the problem cannot be solved. – Jyrki Lahtonen Sep 08 '11 at 09:44
  • @joriki I understand the problem now. I have edited the question. – maxpower Sep 08 '11 at 17:26
  • @Jyriki I understand the problem now. I have edited the question. – maxpower Sep 08 '11 at 17:26
  • @maxpower: Now it's just a matter of making the polynomials match. Substituting $d_i$, $b_i$ and $\gamma$, you get two different expressions for $c$ as sums of products $\alpha^j\beta^i$. One of them contains higher powers than the other, and you can reduce those using the minimal polynomials for $\alpha$ and $\beta$; then you can determine the $c_i$ by solving the resulting system of linear equations. Is this what you're looking for, or are you asking how to use the field structure to do this more efficiently? If you want I can spell this out in more detail later in the evening. – joriki Sep 08 '11 at 17:55
  • @joriki If you know of a method to exploit the field structure to do this more efficiently, that would be great. If not, the solve-a-system-of-linear-equations method is good enough for me. – maxpower Sep 08 '11 at 18:50

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The $c_i$ are not determined by the given data. Different values of $\gamma$ can lead to identical $A(\gamma)$ and $m(\gamma,A)$ but different values of $c_i$, as in the examples given in comments by Jyrki and me.

joriki
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    Wait a minute - you mean, Jyrki isn't just an alternate spelling of joriki? – Gerry Myerson Sep 08 '11 at 11:40
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    @Gerry: No, one is Finnish and the other is Japanese :-) – joriki Sep 08 '11 at 11:45
  • :-) :-) I don't know, whether the pronounciations would be similar though? My first name shares the first syllable with German 'Jürgen', and (according to my experience) is unpronouncable by people whose exposure is to English/Spanish/French only. Given that joriki is a Japanese name I may need to revise the way I mentally pronounce it . – Jyrki Lahtonen Sep 08 '11 at 12:01
  • @Jyrki: It's not a Japanese name; it's a Japanese word that I use as a name :-) (As the White Knight might say, "that's what the name is called" :-) Here's what it means: http://the-wanderling.com/joriki.html. – joriki Sep 08 '11 at 12:08
  • @joriki: Thanks. Sounds interesting! Doesn't really help me to pronounce it, though :-)? Other than that I think I know how the 'ki'-syllable is pronounced. – Jyrki Lahtonen Sep 08 '11 at 12:23
  • @Jyrki: The 'j' is pronounced as in English; the 'r' is sort of half-way between 'r' and 'l' in German/French, and 'o' and 'i' are like in most European languages. – joriki Sep 08 '11 at 12:50