If you keep a, b and c fixed, log a/log b is well approximated by m/l. Making that a linear form in logarithms, that should force l and m to be fairly big. Which is less plausible than solutions at random for small values. This is an exponential Diophantine equation, but not hopeless.

It might be more reasonable to ask this in a more definite context, such as Fredholm theory. The operators considered in the abstract theory of integral equations, for a given class of kernels, are very different in nature from differential operators. But the two theories are related, in some cases, by a type of inversion. You may be asking the question "how extensive is that relationship"?

Try formulating in terms of the Hamming distance, as used in coding theory. en.wikipedia.org/wiki/Hamming_distance . Writing strings of 0s and 1s (with just three 1s) differing in all but two places, is probably clearer. You presumably want an example of such a "code".

Remarks: Van der Waerden's treatment in early editions of his Modern Algebra doesn't fit the pattern: uses primitive elements (perhaps to be constructive, as he notes in his intro that he wants to be). The Dedekind determinant issue does appear to be close enough to linear independence of 1-D characters of G in K*: abelianise G, wlog, and then if characters are dependent the group determinant can't have the basis of eigenvectors we know. Clear to Artin, doubtless.

Well, I'm now confused as to whether what Dedekind proved was morally the evaluation of the "Dedekind determinant", or not, so I'd better adjourn my commentary.

Yes; and see math.uconn.edu/~kconrad/math316/linearchar.pdf for the attribution of the use of the independence in the proof to Artin (with criticism, too). I'm reminded that the Oxford course on Galois theory was or is 16 lectures, the Cambridge one was or is 24 lectures. One clear difference was or is whether separability is treated seriously, so that for example one can give an example of a finite extension without a primitive element.

You can't. But the theory has lasted since the 1930s in the current form.

Is that right? I believe that the independence of multiplicative characters was an innovation. It would have replaced an explicit calculation of determinants, which would be group determinants, which would have been known about in principle since Frobenius ... Anyway that locates the part of the proof of the fundamental theorem where something had to happen (Kaplansky showed that a relatively small amount of something serious proves your adjunction a duality). Speaking of primitive elements, the tacit assumption that extensions are separable would have been a feature before Steinitz?

The answer that comes as "classifying topos of local rings" will doubtless be resisted by geometers; but see Mac Lane-Moerdijk Ch. 8 for all that. What I was alluding to can be traced in Johnstone, Topos Theory, from the index entry on "local ring". Basically adding in "with local ring homomorphisms", in passing from commutative rings to commutative local rings, is a step with general categorical meaning. But the work of Julian Cole referenced there seems ultimately unpublished. (Be careful what you wish for in the "purely categorical"!) Please can we have more expositions of SGA?