There are a few more points I forgot to emphasize in class, even though they’re written in the notes.

The zeroth fact is that is a unit if and only if is non-zero in . This is easy to check, and I leave it to you.

The first fact is that a series

is convergent in if and only if . Thus convergence is very easy to test in . This is one among many aspects of that are much easier than reals. So, for example, when we proved that every element of can be written uniquely as

for , the convergence, that is, the fact that the series defines an element of , is essentially trivial.

Another point regarding the valuation is that while

is always true, in fact, if the two valuations are different, , then we have equality:

This is easy to see if one is zero, so we assume both are non-zero. Then we write

and

with units.

If the valuations are different, then we may assume, without loss of generality, that . But this is just saying that . Now,

But , so is also a unit. Hence,

What might happen to the sum if the valuations are the same?

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