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Let $k$ be a field and $A$ be a central simple algebra over $k$. It's known that $A$ has a splitting field (i.e. a field $K/k$ such that $A_K\cong M_n(K)$ for some $n$) which is finite and Galois.

This allows us to define a reduced norm $N:A\to k$ which is given by the determinant $M_n(K)\to K$ and then descended (via Galois descent) to $k$.

I wonder if we can do the same thing using fpqc descent but avoiding the need for the existence of a finite Galois splitting field, and using only that $\overline{k}$ splits $A$, which is way simpler. (Of course that's not for didactic reasons, since we're exchanging a hard theorem for a harder one. That's just for my curiosity.)

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  • $\begingroup$ A finite index subgroup of $\operatorname{Gal}(\bar k/k)$ fixes $A_{\bar k}$, what else can be said? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 29, 2022 at 13:59
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    $\begingroup$ @მამუკაჯიბლაძე The way I understood the question, it's about the difference between separable and algebraic closure, not about finiteness of the splitting field extension. $\endgroup$
    – Piotr Achinger
    Commented Jan 29, 2022 at 16:41
  • $\begingroup$ @PiotrAchinger precisely! The finiteness is very simple (an isomorphism $A_{\bar{k})\cong M_n(\bar{k})$ involves only a finite amount of data, so we can find a finite extension which does the job). The hard part is proving the existence of a separable splitting field. $\endgroup$
    – Gabriel
    Commented Jan 29, 2022 at 16:48
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    $\begingroup$ You can define the reduced norm using fpqc descent. Have a look at chapter III, section 1.2 in Knus' "Quadratic and Hermitian Forms over Rings", for instance. $\endgroup$
    – Uriya First
    Commented Jan 29, 2022 at 20:09
  • $\begingroup$ @UriyaFirst This solves my question. Thank you. If you would like to turn this into an answer, I'll gladly accept it. $\endgroup$
    – Gabriel
    Commented Jan 30, 2022 at 8:52

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You can define the reduced norm using fpqc descent. Have a look at chapter III, section 1.2 in Knus' "Quadratic and Hermitian Forms over Rings", for instance.

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  • $\begingroup$ Thank you for this reference! The review mentions that this is a relatively young theory, with few books; but Knus's book also is over 30 years old. Is it still the best reference for someone trying to learn the subject, or are there more modern references? $\endgroup$
    – LSpice
    Commented Jan 31, 2022 at 16:19
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    $\begingroup$ @LSpice If you want quadratic forms over rings, then Knus' book is still the best source in my opinion, although the notation has shifted a little since the book was written. There is also a much-shorter-by-comparison monograph by Paul Balmer titled "Witt groups" which gives (tersely) even more modern flavors of the theory of quadratic forms over schemes. Both texts are somewhat advanced, however. Over fields, the most modern reference I am aware of is Elman, Karpenko and Merkurjev's book on the subject: "The Algebraic and Geometric Theory of Quadratic Forms". $\endgroup$
    – Uriya First
    Commented Feb 1, 2022 at 10:51
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    $\begingroup$ @LSpice Continuation of my comment: Scharlau's book "Quadratic and Hermitian Forms" is a little older than Knus' book, focuses mainly on the field case, but is more elementary, and may be better suited as a first read on quadratic forms. It also has a lot of material on local and global fields (if that is your focus), and Chapter 7 in that book may serve as a neat introduction to hermitian forms over rings with involution. Where to start really depends on one's background and goals (and might be the topic of an MO question?). $\endgroup$
    – Uriya First
    Commented Feb 1, 2022 at 11:04
  • $\begingroup$ @‍UriyaFirst, thank you very much! I did not know about the Balmer or EKM book. I am somewhat familiar with Scharlau's book (though I find the treatment of Hermitian forms a bit befuddling), and was wondering where to go next. I will see if I can form a well defined MO question some time. $\endgroup$
    – LSpice
    Commented Feb 1, 2022 at 12:59
  • $\begingroup$ References from @‍UriyaFirst: Balmer - Witt groups; Elman, Karpenko, and Merkurjev - The algebraic and geometric theory of quadratic forms; Scharlau - Quadratic and Hermitian forms. $\endgroup$
    – LSpice
    Commented Feb 1, 2022 at 22:40

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