By Thomas Muir, revised by W.H. Metzler
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Additional info for A Treatise on the Theory of Determinants
The next problems are intended to help with that goal. 5This is a kind of logical skulduggery which delights mathematicians. Look closely at the definition, and you'll see it's correct. 3 Topology m 00 35 but not this! These are allowed ... 2: Balls for non-archimedian absolute values =° Problem 49 Describe the closed ball of radius 1 around the point x in «Jl with respect to the p-adic absolute value. Describe the open ball of radius 1 around x 3; which integers belong to this ball? = Problem 50 Let Ik = «Jl and II = lip.
Show that with respect to this absolute value every open ball is the disjoint union of open balls. ) Do you think this is true for any field with a non-archimedian absolute value? If not, can you come up with a counter-example? 4 37 Algebra So far, we have mostly concentrated on the geometry we obtain from an absolute value on a field Ik. In this section, we take a more algebraic pointof-view, and look for connections between (non-archimedian) absolute values and the algebraic structure 6 of the underlying field.
Now, since (xn) does not tend to zero and is a Cauchy sequence, it must "eventually" be away from zero, that is, there must exist a number c > 0 and an integer N such that IXnl 2: c > 0 whenever n 2: N. ) Now in particular this means that Xn f: 0 for n 2: N, so that we may define a new sequence (Yn) by setting Yn = 0 if n < Nand Yn = l/xn if n 2: N. The first thing to check is that (Yn) is a Cauchy sequence. But that is clear because if n 2: N we have which shows (Yn) E e because II is non-archimedian.
A Treatise on the Theory of Determinants by Thomas Muir, revised by W.H. Metzler