Of course you do. And you are right! Unique factorization of ideals + h-th power is principal. Spot on.

do you mean "unique up to order and units" nontrivial product?

I was not familiar with Bhargava or his thesis work, but I must say, as I am reading his series of papers on composition laws, that it is awesome! So glad I posted this question...

Usually p=2,3 mod 4 refers to a positive prime, and it really does seem that that is what Ben meant, in which case, negative of a prime 3 mod 4 is fine. Hence my comment, as well as Pete's.

this sounds real cool, I didn't know of the relation to the 3 part... Btw, I mention in my question that positive and negative is fine, so negative 3 mod 4 is also fine.

This is a great question. Can someone please post a non-elementary example of a Shimura variety of some CM field and a related F that splits at primes with certain modular congruences? I have never seen an example that isn't a quadratic field of class number 1, or a cyclotomic field.

How about taking the Weil restriction to the rationals of an elliptic curve over a galois field with galois group S_6?

This sounds very interesting.

It is a non a trivial fact that the torsion points on an elliptic curve in Weierstrass form have integer coordinates. If the curve is not in the Weierstrass form, it can have rational torsion points that are not integral. I suggest reading Washington's "Elliptic Curves: Number Theory and Cryptography". It is very detailed and well written.

It is not: p+p^2*Zp is a non-countable subset of the one mentioned by Scott.

@Pete: There does not exist such a sequence. A non-surjective polynomial of degree n over a finite field of q elements has image smaller than q-(q-1)/n. If you bound n and m, then for all but a finite number of finite fields, the bound is smaller than q-m. This was proved by D. Wan in "A p-adic lifting lemma and its applications to permutation polynomials". I found this immediately in the second result of the google search "size of polynomial finite fields".

1) From comments below you'll see you are not the first or even second to propose trying to build such a construction. 2) Other answers certainly do address this reason for a polynomial not working over the reals. 3) Restating the problem is not an answer.

Is Fp a finite field as well? Then no, there does not exist an infinite sequence of finite fields Fq of same characteristic such that Fq is smaller in cardinality than Fp. So T itself is finite, and the field constructed will not be infinite.