Abstract
In this chapter as in Chap. 3, we write K for an arbitrary commutative ring with unit and \(\mathbb{k}\) for an arbitrary field.
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Notes
- 1.
- 2.
By definition, we put \(q^{0}\stackrel{\text{def}}{=}1\).
- 3.
See Sect. 2.8 on p. 35.
- 4.
It will be defined explicitly in Sect. 4.3 on p. 82.
- 5.
Such a factorization is always possible (at least in theory) over an algebraically closed field \(\mathbb{k}\) (see Sect. 3.4.3 on p. 55).
- 6.
See formula (4.8) on p. 79.
- 7.
Do not worry if you are not conversant with some of them. We will deduce them all soon.
- 8.
Jacob Bernoulli (1654–1705) did this job in about seven minutes having just pen and paper, as he wrote (not without some pride) in his Ars Conjectandi, published posthumously in 1713 (see [Be]).
- 9.
To begin with, I recommend Chapter 15 of the book A Classical Introduction to Modern Number Theory, by K. Ireland and M. Rosen [IR] and Section V.8 in the book Number Theory by Z. I. Borevich and I. R. Shafarevich [BS]. At http://www.bernoulli.org/ you may find a fast computer program that evaluates B 2k as rational simplified fractions.
- 10.
Compare with the proof of Proposition 4.1 on p. 76.
- 11.
Here we use that n = 1 + ⋯ + 1 ≠ 0 in \(\mathbb{k}\).
- 12.
Here we use again that \(\mathop{\mathrm{char}}\nolimits \mathbb{k} = 0\).
- 13.
Note that the exponent of x grows along the horizontal axis.
- 14.
But not all, in general.
- 15.
Note that this gives another proof of Lemma 4.5.
- 16.
See Proposition 4.1 on p. 76.
- 17.
That is, consisting of n cells; the number p(n) is also called the nth partition number; see Example 1.3 on p. 6.
- 18.
Compare with Problem 3.12 on p. 67.
- 19.
Compare with Example 4.7 on p. 89.
- 20.
The map \(F: \mathbb{Q}[t] \rightarrow \mathbb{Q}[t]\) is linear if for all \(\alpha,\beta \in \mathbb{Q}\) and all \(f,g \in \mathbb{Q}[t]\), the equality F(α ⋅ f +β ⋅ g) = α F( f) +β F(g) holds. For example, all the maps \(D = \frac{d} {dt}\), D k, \(\Phi (D)\) for all \(\Phi \in \mathbb{Q}[\!\![ x]\!\!] \), ∇, ∇k, and \(\Phi (\nabla )\) are linear.
References
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Borevich, Z. I., Shafarevich, I. R.: Number Theory. Academic Press, New York (1966).
Danilov, V. I., Shokurov, V. V.: “Algebraic Curves, Algebraic Manifolds and Schemes.” In Encyclopedia of Mathematical Sciences. Springer, Heidelberg (1994).
Francis, G. K.: A Topological Picturebook. Springer, Heidelberg (1987).
Gorenstein, D., Lyons, R. Solomon, R.: The Classification of the Finite Simple Groups. Mathematical Surveys and Monographs 40, vols. 1–6. AMS Providence, R.I. (1994–2005).
Humphreys, J. E.: Linear Algebraic Groups. Springer, Heidelberg (1975).
Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. Springer, Heidelberg (1990).
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Gorodentsev, A.L. (2016). Elementary Functions and Power Series Expansions. In: Algebra I. Springer, Cham. https://doi.org/10.1007/978-3-319-45285-2_4
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