I am trying to learn about $p$-adic numbers and I have some questions.
I tried to learn about the motivation from the Wikipedia page, which was very clear and with some straightforward examples.
But I'm having trouble understanding the following:
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If we take an algebraic approach to construction, we first construct $\mathbb{Z}_p$ and them from it we construct $\mathbb{Q}_p$ by taking it to be the field of fractions of $\mathbb{Z}_p$. The thing I don't understand is the zero in the ring $\mathbb{Z}_p$. For example, if we have $p=7$ we have that (0,7,7,7,7,…) and (0,7,7,7,56,56,56,….) both have no inverses. Are they both zero in $\mathbb{Z}_p$?
Is this the same thing as 7 and 14 being both zero in $\mathbb{Z}/7\mathbb{Z}$? -
Every $x\in \mathbb{Q}_p$ can be written as $x=p^{-n}u,$ where $n\in \mathbb{N}, u \in \mathbb{Z}_p^*.$ Why is this true? So I understand that $x=\frac{a}{b}, a,b\in \mathbb{Z}_p,$ so we have $a=p^ku, b=p^mv, m,k\in\mathbb{N}, u,v\in \mathbb{Z}_p.$ But I don't know how to continue.
Best Answer
no they are not zero.it doesn't have an inverse in $\mathbb{Z}_7$ but this doesn't mean it should be zero $\mathbb{Z}_7$ is a ring like $\mathbb{Z}$ and has many non-zero,non-invertible elements.
we can describe invertible elements in $\mathbb{Z}_7$: a sequence is invertible in $\mathbb{Z}_7$ if and only if it's first component is not divisible by seven. the reason is that every element of $\mathbb{Z}/7^n$ which is coprime with 7 has a unique inverse. so you can take the inverses componentwise and they are compatible with each other.
by this fact you can see that every element of $\mathbb{Z}_7$ can be written in the form $7^nu$ where u is invertible.now your second question is this fact in another language.