Abstract
In this chapter, we review some facts from abstract algebra. Since we assume that the reader is familiar with basic abstract algebra, we put the emphasis on the concepts that are particularly important in this book, such as the ring of polynomials, modules over this ring, algebraic and inseparable field extensions, finite fields, and central simple algebras.
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Notes
- 1.
Unfortunately, for now, Conrad’s notes exist only electronically, with a separate file for each topic, and the notes are revised occasionally, which makes making precise references to them difficult.
- 2.
A notable exception is the case when R is complete with respect to a metric and there is an analytic notion of convergence with respect to that metric, e.g., ∑n≥0 x n∕n! can be evaluated at an arbitrary \(c\in \mathbb {C}\). We will return to this in Chap. 2.
- 3.
Consider f(x) = x 2 − 1 in \(\mathbb {Z}/8\mathbb {Z}[x]\). Then 1, 3, 5, 7 are roots of f(x), so f(x) has 4 roots even though it has degree 2. This, of course, does not contradict the claim, as \(\mathbb {Z}/8\mathbb {Z}\) is not a field. Also note that in \(\mathbb {Z}/6\mathbb {Z}[x]\), we have the factorization x = (3x + 4)(4x + 3). Since neither of these factors is a unit, x is reducible in \(\mathbb {Z}/6\mathbb {Z}[x]\).
- 4.
In general, M tor need not be a submodule of M, and even if it is a submodule, there might not exist another submodule M′⊂ M such that M≅M′⊕ M tor; see Exercise 1.2.1.
- 5.
But note that if L∕E is normal and E∕K is normal, then L∕K is not necessarily normal; for example, \(\mathbb {Q}\subset \mathbb {Q}(\sqrt {2})\subset \mathbb {Q}(\sqrt [4]{2})\). Hence, a tower of Galois extensions is not necessarily Galois.
- 6.
Recall that in product topology the open sets in ∏i ∈ I G i are the unions of sets of the form ∏i ∈ I U i, where U i ⊆ G i and U i ≠ G i for only finitely many i.
- 7.
With respect to this topology, \(\varprojlim G_i\) is compact and totally disconnected.
- 8.
If F is a finite field of characteristic p, then x n − 1 cannot split completely over F if n > #F and \(p\nmid n\).
- 9.
Sometimes the notation K m is used for a vector space of dimension m over K, or for the direct product of m copies of K. Here \(K^{p^n}\) has a different meaning.
- 10.
This equivalent definition applies to any field, not necessarily of positive characteristic, so a field of characteristic 0 is perfect.
- 11.
Recall that, given two functions \(f, g\colon \mathbb {R}_{>0}\to \mathbb {R}_{>0}\), the notation f(x) = o(g(x)) means that f(x) ≤ M ⋅ g(x) for any M > 0 as x →∞, and f(x) = O(g(x)) means that f(x) ≤ M ⋅ g(x) for some M > 0 as x →∞.
References
Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976.
A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966, Translated from the Russian by Newcomb Greenleaf.
Keith Conrad, Expository papers, https://kconrad.math.uconn.edu/blurbs/.
David S. Dummit and Richard M. Foote, Abstract algebra, third ed., John Wiley & Sons, Inc., Hoboken, NJ, 2004.
Harley Flanders, The norm function of an algebraic field extension, Pacific J. Math. 3 (1953), 103–113.
_________ , The norm function of an algebraic field extension. II, Pacific J. Math. 5 (1955), 519–528.
_________ , Algebra, third ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002.
Rudolf Lidl and Harald Niederreiter, Introduction to finite fields and their applications, first ed., Cambridge University Press, Cambridge, 1994.
Daniel A. Marcus, Number fields, Universitext, Springer, Cham, 2018, Second edition of [MR0457396], With a foreword by Barry Mazur.
James S. Milne, Fields and Galois theory (v4.30), 2012, Available at www.jmilne.org/, p. 124.
_________ , Class field theory (v4.02), 2013, Available at www.jmilne.org, pp. 281+viii.
David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970.
I. Reiner, Maximal orders, London Mathematical Society Monographs. New Series, vol. 28, The Clarendon Press, Oxford University Press, Oxford, 2003, Corrected reprint of the 1975 original, With a foreword by M. J. Taylor.
Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980.
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Papikian, M. (2023). Algebraic Preliminaries. In: Drinfeld Modules. Graduate Texts in Mathematics, vol 296. Springer, Cham. https://doi.org/10.1007/978-3-031-19707-9_1
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