To be honest, I've never see the injectivity of the transfer map emphasized in such a foundational role. It's probably equivalent to things that are (to me) much more familiar -- the Neukirch's "class field axiom" seems likely to be equivalent (though I don't have any references nearby) since they're both described cohomologically. In fact, the equivalence might be seen directly from the Hochschild-Serre spectral sequence, though again, this is pure speculation. It is exactly this class field axiom that makes abstract class field theory work so this might be exactly what you're looking for.

Just a comment that you can certainly improve the efficiency of your brute force approach. For example, once you have one determinant calculated, you may as well cross off all matrices which are row or column swaps. While we're at it, the 6x6 determinants properly include into the 7x7 determinants (multiple times!), so you could skip over these if you've already done them. And I'm sure you could be much more clever than these...this was just a first thought.

Thanks for the tips, everyone. Also, Haberland's book seems largely subsumed by Koch's "Galois Theory of p-Extensions" (for which Franz has written an excellent postscript), and anything missing from there is almost surely in NSW. But boy is it hard to get hold of.

Sure thing. Nice question.

Well, cycling well-viewed topics back to the front comes at the cost of pushing newer questions out of immediate visibility faster, so I understand the motivation. On the other hand, as the site grows, we get new perspectives on old questions which, and as vonjd points out, are apparently still of interest. We shouldn't close things just because the site old-timers are tired of seeing them. This discussion is probably on meta somewhere already....

I only regret that I have but one +1 to give to this entry.

Aha, thanks. For anyone else who's looking for the link, it's part 3 (pages 117-141) of: Martin, W. T.; Chern, S. S.; Zariski, Oscar. Scientific report on the Second Summer Institute, several complex variables. Bull. Amer. Math. Soc. 62 (1956), 79--141.

Is this the Zariski paper you're referencing? Sounds fantastic. The fundamental ideas of abstract algebraic geometry. Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, 77--89. Amer. Math. Soc., Providence, R. I., 1952.