By Engstrom H. T.
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Example text
1. The reader is encouraged to work out and keep in mind such restatements in the case of p-adic absolute values j jp and the corresponding valuations wp . Incidentally, the valuation group of wp is ޚ. 2. We now fix a field K and an absolute value j j on K. Definition 6. an /n in K is called a Cauchy sequence (with respect to j j, or a j j-Cauchy sequence) if, for every real number " > 0, there exists N 2 ގ such that jan am j < " for all m; n > N: If every j j-Cauchy sequence in K converges with respect to j j to an element of K, we say that K is complete (with respect to j j).
M C 1/jajm : Now taking the m-th root and the limit as m ! jaj; jbj/; which is (iii). jaj ; jbj / to show that j j satisfies the strong triangle inequality. That it satisfies the first two defining properties of an absolute value is obvious regardless of (iii). Finally, assume (iv) and take n 2 ގ. We know that j j m is an absolute value for any m 2 ; ގhence jnjm Ä n by the triangle inequality. Taking the m-th root and the limit as m ! 1 yields jnj Ä 1, proving (i). Remark. Let p be a prime number.
K dim V ) to a diagonal form: (3) q ' Œa1 ; : : : ; am : In the sequel we will tacitly assume all quadratic forms to be nondegenerate. This means that the entries ai in (3) all lie in K . V; q/ ? V ˚ V 0 ; q ? q ? x 0 ; y 0 / 30 22 Orders and Quadratic Forms for x; y 2 V and x 0 ; y 0 2 V 0 . With the standard identifications K m ˚ K n D K mCn and K m ˝ K n D K mn , we therefore have (6) Œa1 ; : : : ; am ? V; q/ is denoted by k Every q possesses an orthogonal decomposition q. q D q0 ? v/ D 0 — and a hyperbolic component q1 , which by definition means q1 ' k H D k Œ1; 1.