By Colin Maclachlan, Alan W. Reid

Lately there was enormous curiosity in constructing suggestions in keeping with quantity thought to assault difficulties of 3-manifolds; includes many examples and plenty of difficulties; Brings jointly a lot of the present literature of Kleinian teams in a transparent and concise approach; at the present no such textual content exists

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**Example text**

If v is non-Archimedean, then v belongs to a place corresponding to a J>rime ideal P. The field kp is usually referred to as a P-adic field. The valuation ring of kp with respect to the extended valuation fJp is the ring of P-adic integers and is denoted by Rp . Recall that the valuation ring ll( vp) of k with respect to Vp is a discrete valuation ring whose unique r naximal ideal is generated by an element 1r E Rk. The same can be proved for the ring Rp. More precisely, the following holds: an The valuation ring Rp of the completion kp is a discrete valuation ring whose unique maximal ideal is generated by ip (1r ) .

I ) . 20 8. 0. 82 , . . , ,Bd } · Prove that N ( I ) 2 = discr{ /h , /h . . 8d } �k . 3. 14 . 9. Let l I k be an abelian extension. Show that there are infinitely many primes P with splitting pattern ([l : k] , 1) if and only if g is cyclic. 10. Show that in any finite extension l I k of number fields, there are infinitely many primes P of k which split completely in l. 1 1 . Deduce from Dirichlet 's density theorem that there are infinitely many rational primes in any arithmetic progression { an + d I ( a, d) = 1 } .

I f k is a number field, then from the preceding section we can embed k in l . l u · completions kv for each finite and infinite place of k. Thus if (V, B) regular quadratic space over k, then it gives rise to regular quadratic :;paees over the local fields, CC , JR, and It> for each finite place P. ltcgular quadratic spaces over CC are classified up to isometry by their di I f 'nsion and over lR by their dimension and signature, which is the number , f positive eigenvalues minus the number of negative eigenvalues.