Question regarding root of Bring quintic not expressible with radicals

[fresh_42]

Your post #33 indicates that the simultaneous equations we are trying to solve in Galois theory are non-linear. This suggests that that the theory of groups and field automorphisms is useful in reasoning about the solutions of special types of non-linear equations.

Yes, you're right. It is basically of the same difficulty as finding a relation in a group, or to show there is none. This is - as far as I know - not a trivial issue. I'm still thinking about @Buzz Bloom 's quest to solve it. At least the isomorphism $\mathbb{Q}(\sigma_i) \cong \mathbb{Q}[x]/(x^5-x+1)$ allows us to only consider polynomials and no fractions. I think the automorphism group is the whole $\mathcal{Sym}(5)$, but I have no idea so far, how to prove the $\sigma_i$ actually behave like variables, or indeterminates, i.e. have no non-trivial relation $f(\sigma_0,\ldots,\sigma_4)=0\,.$

 

[mathwonk]

one year after teaching graduate algebra many times without getting to galois theory, i wanted to understand it so much that I began my course with this topic. It turned out one cannot do that, since one needs to know about polynomial rings, vector space dimension, and groups, including the concept of normal sub groups and simple groups. After developing this prerequisite material, I did address the problem of which polynomials can be solved by radicals, in detail, in section 843-2 of my algebra course notes. I tried to survey this in the first 7 pages or so of these notes, and then spent the rest of the 56 or so pages filling in the details. See if reading the first 4, or first 7 pages of these free notes helps: If necessary you may want to refer back to the notes from 843-1 for background on groups.

http://alpha.math.uga.edu/%7Eroy/843-2.pdf

If you are still interested in why X^5 - X + 1 has no root expressible over Q by radicals, note that if it did, then taking the negative of that root would give such a root of X^5 -X -1, and this is a standard example of an equation with non solvable Galois group, going back to the book of Van der Waerden and reproduced in some more modern books like Dummit and Foote. Also see these free notes, example 3.33. The technique is to reduce the equation modulo the primes 2,3, and use the fact that those reduced Galois groups are isomorphic to subgroups of this one, and in a way that preserves the cycle type of the permutations of the roots.

http://citeseerx.ist.psu.edu/viewdoc/download?rep=rep1&type=pdf&doi=10.1.1.211.2314

pages 16 and 17 of those notes are especially relevant to your question, and explain how to check in many cases, using a computer, whether a galois group is isomorphic to S(p) or A(p).

 

[Stephen Tashi]

It turned out one cannot do that, since one needs to know about polynomial rings, vector space dimension, and groups, including the concept of normal sub groups and simple groups.

Even after those topics in abstract algebra are understood there is still a big pedagogical problem of explaining the connection between the definition of "solving an equation by radials" in the sense it is understood in secondary school algebra and the re-definition of that concept which is used in abstract algebra. I wonder if the solution to this pedagogical problem is to employ even more abstraction by applying mathematical ideas from the study of abstract formal languages -i.e. formal rules for manipulating strings of symbols. (Something like what Ritt did for the concept of "closed form solutions" https://en.wikipedia.org/wiki/Joseph_Ritt .)

For example, considering Abel's proof, as expounded on a blog: http://fermatslasttheorem.blogspot.com/2008/09/abels-impossibility-proof-radicals-of.html , we find that it uses language that could be rephrased as properties of string manipulations in some formal computer language. It uses ideas such as "nested" , "at the deepest level" , "of the form" etc.

e.g.

If the solution to the general quintic is expressible as nested radicals then for all radicals of the form $R^{1/m}$, $m = 2$.

e.g. ( http://fermatslasttheorem.blogspot.com/2008/10/abels-impossibility-proof.html)

If a solution to this equation exists it can be expressed as follows (see Theorem 5 here):

$y = p + R^{1/m} + p_2 R^{2/m} + p_3 R^{3/m} + ...+ p_{m-1} R^{(m-1)/m}$

where $m$ is a prime number and $R,p_1,p_2..$ .are functions of this same form finitely nested at the deepest level each $p,p_i, R$ is a function of the coefficients of the general quintic equation.

From the point of view of intellectual honesty, to formally present the connection between the elementary algebra sense of solution by radicals and the abstract algebra definition of the concept, it is necessary to formalize all the language involving concepts of string manipulations (e.g. "nested" , "of the same form"). To do this in the typical algebra course would be a kind of mathematical culture shock because the study of formal languages is traditionally the domain of the Computer Science department.

Traditional graduate education of algebraists can proceed satisfactorily by hand-waving and citing the typical examples (e.g. quadratic and cubic equations) to connect the elementary algebra concept of solution by radicals to its abstract re-definition. I'm not advocating a revision to the standard curriculum. However, for the benefit of people who are curious about the connection between the two concepts and also understand formal presentations, it would be nice to have a formal presentation of string manipulation aspects of the topic.