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In Galois theory, the endeavor of algorithmically finding the Galois group of a polynomial $f$ over the field $\mathbb{Q}$, where $f\in\mathbb{Q}[z]$ is generally not well-accomplished. By this, i mean that the algorithms tend to have caveats like: computability exceptions, large complexity values, not covering the entire $\mathbb{Q}[z]$, and so on. I know that there must be algorithms that do a very good job, but my question isn’t about the algorithms per se, but about why these algorithms tend to have these problems. If all Galois theorems apply, and are perfectly capable of constructing a solid theory that describes the Galois group of any given polynomial, shouldn’t it be a relatively easy task to implement such an algorithm? As an alternative question; is the constructions of these algorithms not entirely straightforward?

Simón Flavio Ibañez
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  • While I know nothing about the case of Galois groups it is not at all unusual for algorithms for simple problems to behave poorly. A classic example would be factorizing integers, it is very straightforward to construct an algorithm to perform it but getting a fast algorithm is much more difficult. There are also cases where special cases of a probllem can be much easier to solve than general cases, I believe the SAT type problems are an example. There are also classes of problems where approximate solutions are easy but exact ones are hard. – Fishbane May 21 '23 at 03:50
  • As well as this there are some problems that are easy to describe and which have well defined answers but which have no algorithm to solve them. The classic example is the halting problem, but there are many other examples, one example which is a little more mathematical is the inability to classify 4 manifolds. – Fishbane May 21 '23 at 04:01
  • What kind of fields $\Bbb F$ are you allowing here? If $\Bbb F$ is allowed to be any field, then the problem of giving the Galois group is not computable. – Lukas Heger May 21 '23 at 04:13
  • @LukasHeger, but that does not really say anything, because there are uncomputable fields... – Mariano Suárez-Álvarez May 21 '23 at 04:13
  • What implementations are you talking about that have al these problems? The fact that the complexity is high is more or less expected, but the rest? – Mariano Suárez-Álvarez May 21 '23 at 04:17
  • @MarianoSuárez-Álvarez true, but even that trivial kind of example shows that you must restrict the allowed fields. But there are even recursive field for which the Galois group problem is not computable. – Lukas Heger May 21 '23 at 04:18
  • @Fishbane I know that some algorithms that do simple tasks have these same problems, but for most we know “why”. As an example, the interger factorization algorithm has to cope with all the possible prime factors a given integer can have, an thus, for large N, it can become a very complex task. – Simón Flavio Ibañez May 21 '23 at 04:50
  • @LukasHeger you’re right. I’m actually most interested in the $\mathbb{Q}$ case. I’ll edit now. – Simón Flavio Ibañez May 21 '23 at 04:52
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    @SimónFlavioIbañez Could one not by analogy say that for any polynomial of order $n$ the algorithm has to cope with all of the groups of order less than $n$ which could be the galois group. This grows very fast so I see no reason why this could not be given as the reason in the same way you use the number of factors as the reason in the integer factorization case. – Fishbane May 21 '23 at 05:05
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    One aspect (even over a nice field like $\Bbb{Q}$) is that, given the degree $n$ of the polynomial, the number of alternative Galois groups blows up on your face (and their classification is anything but straight forward). So the algorithm needs to distinguish between an astronomical number of cases sooner rather than later. – Jyrki Lahtonen May 21 '23 at 05:21
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    FWIW, a bright advanced undergrad student of mine tried to implement an algorithm based on the idea outlined by David E. Speyer here, using our ability to calculate the (complex) roots of a polynomial in $\Bbb{Q}[x]$ to very high precision. He ran into exactly the problem of trying to identify the group (if I correctly identified the issues he later asked about). – Jyrki Lahtonen May 21 '23 at 05:27
  • @Fishbane Sorry about duplicating your last point. Don't know how I missed it in my first reading of the comment chains. I was preparing my own for a little while :-( – Jyrki Lahtonen May 21 '23 at 05:33
  • @JyrkiLahtonen Don't worry about it, I'm not really specialized into Galois theory so it is good to have statements from someone who knows more about this specific case. – Fishbane May 21 '23 at 05:37
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    But "identifying" (I don't even know what that means, really!) the group is an entirely different problem, no? Computing the Galois group should content itself with giving a generating set of permutations, for example. – Mariano Suárez-Álvarez May 21 '23 at 07:55

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