2

Assuming $p$ is a prime obviously, am I missing something or are both sets just $\{0, 1, \dots, p-1\}$.

I also still don't fully understand what a quotient is, so maybe that has something to do with this? Please enlighten me.

qualcuno
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Idk Dk
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    What's your definition of $\mathbb Z_p$? The notation is used for different rings, depending on context. – Christoph Feb 11 '21 at 15:16
  • @Christoph i honestly am not sure, ive seen it in 2 courses, i think it vaguely means the set i mentioned above – Idk Dk Feb 11 '21 at 15:18
  • The notation $\mathbb Z/p \mathbb Z$ is standard for integers modulo $p$, as a set one can think of it as equivalence classes ${[0],\ldots, [p-1]}$ or as you say, as numbers from $0$ to $p-1$ with appropriate multiplication/product. The notation $\mathbb Z_p$ has two standard meanings: one is the so called "topologists notation", $\mathbb Z_p = \mathbb Z/p \mathbb Z$. The other one refers to the $p$-adic integers. Which one you are working with depends on the context. – qualcuno Feb 11 '21 at 15:18
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    Personally, I try to reserve the notation $\mathbb Z_p$ for the p-adic integers but that rule isn't universally followed. – lulu Feb 11 '21 at 15:19
  • @guidoar so is $\mathbb{Z}/p\mathbb{Z}$ just the set of all integers modulo $p$? For example, $p + 1$ is just 1 in this set? – Idk Dk Feb 11 '21 at 15:21
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    Yes, $p+1=1$ in $\Bbb Z/p\Bbb Z$, because $p=0$. Actually, this is written as $\Bbb F_p$, the finite field with $p$ elements. Here we can compute like usual, because it is a field (only a bit different, that $p=1+1+\cdot +1=0$). So it is not just "in this set", but really in a field. – Dietrich Burde Feb 11 '21 at 15:23
  • @guidoar Where did you hear "topologists notation"? In my experience $,\Bbb Z_p := \Bbb Z/p,$ is far more common in introductory number theory (and discrete mathematics), i.e. in contexts where $p$-adics are not used so there is no chance of notation clash. – Bill Dubuque Feb 11 '21 at 19:30
  • This term is (informally) used for example in the 'Notational Conventions' section of Brown's Cohomology of Groups. I remember reading it on this site a couple times too. I don't know the origin of this terminology, though (and I agree with your comment) – qualcuno Feb 11 '21 at 19:36
  • @BillDubuque I have just found a historical remark along the lines of this usage on Warning 0.0.5 of May and Ponto's More Concise Algebraic Topology. – qualcuno Feb 11 '21 at 19:40
  • Namely: "We warn the reader that algebraic notations in the literature of algebraic topology have drifted over time and are quite inconsistent. The reader may find $\mathbb Z_p$ used for either our $\mathbb Z_{(p)}$ or for our $\mathbb F_p$; the latter choice is used ubiquitously in the “early” literature, including most of the first author’s papers...." – qualcuno Feb 11 '21 at 19:42
  • (cont.) "...In fact, regrettably, we must warn the reader that $\mathbb Z_p$ means $\mathbb F_p$ in the book [91]. The $p$-adic integers only began to be used in algebraic topology in the 1970’s, and old habits die hard. In both the algebraic and topological literature, the ring $\mathbb Z_p$ is sometimes denoted $\widehat{\mathbb Z}_p$; we would prefer that notation as a matter of logic, but the notation $\mathbb Z_p$ has by now become quite standard". – qualcuno Feb 11 '21 at 19:42
  • @guidoar Interesting, thanks much for the links. In case this thread gets deleted you might want to move the comments somewhere more stable, e.g. the linked dupe, though there may be a better place. – Bill Dubuque Feb 11 '21 at 20:21

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