The easiest to understand line by line are the elementary proofs that go through Gauss' Lemma, and are likely to be seen in any elementary number theory book. I've never actually liked these proofs personally and prefer the one at the start of Serre's "A Course in Arithmetic" for a proof without many technical prerequisites (finite fields only from memory).

Edit: Less elementarily, one could argue that any 'best' proof will necessarily be adelic, proving that the product of the Hilbert symbols over all places at rational arguments is 1. For a geometric example of such an argument, I'm going to throw out arXiv:0804.2142 and references therein, primarily because I don't know if there exists an analogous argument using a central extension of GL_1 in the number field case.

The easiest to understand line by line are the elementary proofs that go through Gauss' Lemma, and are likely to be seen in any elementary number theory book. I've never actually liked these proofs personally and prefer the one at the start of Serre's "A Course in Arithmetic" for a proof without many technical prerequisites (finite fields only from memory).

Edit: Less elementarily, one could argue that any 'best' proof will necessarily be adelic, proving that the product of the Hilbert symbols over all places at rational arguments is 1. For a geometric example of such an argument, I'm going to throw out arXiv:0804.2142 and references therein, primarily because I don't know if there exists an analogous argument using a central extension of GL_1 in the number field case.

The easiest to understand line by line are the elementary proofs that go through Gauss' Lemma, and are likely to be seen in any elementary number theory book. I've never actually liked these proofs personally and prefer the one at the start of Serre's "A Course in Arithmetic" for a proof without many technical prerequisites (finite fields only from memory).

The easiest to understand line by line are the elementary proofs that go through Gauss' Lemma, and are likely to be seen in any elementary number theory book. I've never actually liked these proofs personally and prefer the one at the start of Serre's "A Course in Arithmetic" for a proof without many technical prerequisites (finite fields only from memory).

Edit: Less elementarily, one could argue that any 'best' proof will necessarily be adelic, proving that the product of the Hilbert symbols over all places at rational arguments is 1. For a geometric example of such an argument, I'm going to throw out arXiv:0804.2142 and references therein, primarily because I don't know if there exists an analogous argument using a central extension of GL_1 in the number field case.