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Could someone explain me the following Lemma 4.3.1 from the book "Field Arithmetic", 3rd edition by Michael D. Fried and Moshe Jarden:

Let $F/K$ be a function field and denote the unique extension of $K$ of degree $r$ by $K_r$. Also, denote the $F_r$ the "extension" of $F$ to $F_r$.

Now the claim is as follows:

Let $\mathfrak p$ be a prime divisor of $F/K$. Then $\mathfrak p$ decomposes in $F_r$ as $\mathfrak p=\mathfrak P_1+\mathfrak P_2+\cdots +\mathfrak P_d$, where $\mathfrak P_1,\ldots,\mathfrak P_d$ are distinct prime divisors of $F_r/K_r$, $\operatorname{deg}(\mathfrak P_i)=r^{-1}\cdot \operatorname{lcm}(r,\deg(\mathfrak p))$ and $d=\operatorname{gcd}(r,\deg(\mathfrak p))$.

I don't see how the authors compute $d$ as they refer to some proposition 2.6 but there is no such a proposition on the book. So how do one proves such a lemma?

Jaakko Seppälä
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  • Assuming $K$ is a finite field (given that the degree $r$ extension is unique) then this follows from the fact that the residue class field $k_{\mathfrak{P}i}$ is the compositum of the residue class field $k{\mathfrak{p}}$ and $K_r$. By Galois theory of finite fields that compositum has degree $\mathop{lcm}(r,\deg(\mathfrak{p}))$. Because after extension by scalars to $K_r$ we count degrees as the extension degrees over $K_r$, we normalize by dividing by $r$. – Jyrki Lahtonen Aug 31 '23 at 16:14
  • Thanks! Indeed, there was an assumption that $K$ is a finite field. – Jaakko Seppälä Aug 31 '23 at 16:17
  • Frantically searching the site for explanations of the simpler version in the case of the rational function field. In that case part of the explanation comes from the fact that $F_r/F$ is Galois. Compare with this oldie or this. Uniqueness of extensions of $K$ of a given degree is the key to the rest: the roots of the factors are roots of the original polynomial, and hence generate the same extension field over the prime field. For more general function fields the argument is more involved. – Jyrki Lahtonen Aug 31 '23 at 16:24
  • Link to the book. – Jyrki Lahtonen Aug 31 '23 at 16:26
  • I'm relatively familiar with Stichtenoth's and Rosen's book on the same theme, so making sense of Fried & Jarden takes a while. I don't see them needing anything from section 2.6, but I also find a cryptic comment along the lines that in contrast to our exposition Stichtenoth always assumes the field of constants to be perfect. Apparently Fried & Jarden want a more general exposition, and functions fields with a single variable can (in selected parts of the book, not in chapter 4 though) have infinite fields of constans. – Jyrki Lahtonen Aug 31 '23 at 16:50
  • The reference to 2.6. does feel a bit strange. The calculation in the proof of 4.3.1. may be needing material from sections 2.3 and 2.4. (my first thought was that the calculation of $d$ is an application of the $n=efg$ formula from there). That may have been rewritten in a language specific to constant field extension somewhere in between. I'm afraid I can't put more time into searching right now :-( – Jyrki Lahtonen Aug 31 '23 at 17:01
  • I asked this by Moshe Jarden. He wrote me that there is a mistake in the book and the result follows from the Proposition 2.3.2 of the same book. But for me this is not obvious. – Jaakko Seppälä Mar 16 '24 at 16:25

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