Polynomials and Simple Field Extensions

Gorodentsev, Alexey L.

doi:10.1007/978-3-319-45285-2_3

Alexey L. Gorodentsev²

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Abstract

In this chapter, K will denote an arbitrary commutative ring with unit and \(\mathbb{k}\) an arbitrary field.

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Notes

1.
Formally speaking, these rules define addition and multiplication of sequences (a _ν), (b _ν) formed by elements of K. The variable x is used only to simplify the visual perception of these operations.
2.
See Sect. 2.8 on p. 35.
3.
See Sect. 2.8.2 on p. 36.
4.
By definition, this means that 1 = h ₁ f ₁ + h ₂ f ₂ + ⋯ + h _n f _n for some \(h_{i} \in \mathbb{k}[x]\) (see Sect. 2.3 on p. 26).
5.
See Sect. 4.2 on p. 76.
6.
With our current equipment it is not so obvious. However, it follows at once from Gauss’s lemma (see Lemma 5.4 on p. 117) by means of an Eisenstein-type argument (see Example 5.8 on p. 119).
7.
This polynomial can be computed explicitly by the Euclidean algorithm applied to F _i and f _i.
8.
Although this fact is widely known as the fundamental theorem of algebra, its whys and wherefores are rather analytic-geometrical.
9.
Also called the modulus or absolute value of z.
10.
That is, taken with a positive sign when an arc goes counterclockwise and with a negative sign otherwise.
11.
See Example 2.4 on p. 22.
12.
That is, the composition of a rotation and dilation with the same center (since they commute, it does not matter which one was done first).
13.
Note that the multiplication in \(\mathbb{R}\) agrees with that defined in (3.19).
14.
An endomorphism ι: X → X of a set X is called an involution if ι ∘ι = Id_X.
15.
In other terminology, a generating root of unity.
16.
This is not so easy, and we shall return to this problem in the forthcoming Algebra II.
17.
See Sect. 3.5.4 on p. 60.
18.
Note that the equality − 1 = 1 in \(\mathbb{F}_{2}\) allows us to avoid the minus sign.
19.
The standard Cartesian coordinates (x, y) in \(\mathbb{C}\) are related to the “triangular coordinates” (u, w) of the same point x + iy = z = u + w ω by x = u − w∕2, \(y = \sqrt{3}w/2\).
20.
We take such a roundabout way because the vector-space machinery will not appear until Chap. 6 Indeed, \(\mathbb{F}\) is a finite-dimensional vector space over \(\mathbb{k} \subset \mathbb{F}\), and Lemma 3.3 follows at once from Corollary 6.3 on p. 133.
21.
See Theorem 2.1 on p. 30.
22.
See Sect. 3.4.2 on p. 54.
23.
Especially for a finite one.
24.
See Problem 2.20 on p. 39.
25.
See Problem 2.20 on p. 39.
26.
Independently of Theorem 3.2 on p. 63.
27.
See Sect. 3.5.4 on p. 60.
28.
For example, a point \(p \in \mathbb{C}\) is the limit of a sequence if every ɛ- neighborhood of p contains all but a finite number of elements of the sequence. A set U is open if for every z ∈ U, some ɛ- neighborhood of z is contained in U. A function \(f: \mathbb{C} \rightarrow \mathbb{C}\) (or function \(g: \mathbb{C} \rightarrow \mathbb{R}\)) is continuous if the preimages of all open sets are open.
29.
Hint: First show that \(\forall \,M \in \mathbb{R}\), \(\exists \,R > 0\): | f(z) | > M as soon as | z | > R.
30.
Recall that a noninvertible element a in a commutative ring with unit is reducible if a = bc for some noninvertible b, c; otherwise, a is irreducible .
31.
See formula (3.22) on p. 65.
32.
See Problem 2.18 on p. 39.

References

Bernoulli, J.: Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis, et epistola gallicé scripta de ludo pilae reticularis. Basileae, impensis Thurnisiorum, fratrum (1713).
Google Scholar
Borevich, Z. I., Shafarevich, I. R.: Number Theory. Academic Press, New York (1966).
Google Scholar
Danilov, V. I., Shokurov, V. V.: “Algebraic Curves, Algebraic Manifolds and Schemes.” In Encyclopedia of Mathematical Sciences. Springer, Heidelberg (1994).
Google Scholar
Francis, G. K.: A Topological Picturebook. Springer, Heidelberg (1987).
Google Scholar
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).
Google Scholar
Humphreys, J. E.: Linear Algebraic Groups. Springer, Heidelberg (1975).
Google Scholar
Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. Springer, Heidelberg (1990).
Google Scholar
Mumford, D.: Tata Lectures on Theta I. Progress in Math, vol. 23, Birkhäuser (1983).
Google Scholar
Serre, J.-P.: Lie Groups and Lie Algebras: 1964 Lectures Given at Harvard University. Lecture Notes in Mathematics series 2, vol. 1500, Springer (1965).
Google Scholar

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Faculty of Mathematics, National Research University “Higher School of Economics”, Moscow, Russia
Alexey L. Gorodentsev

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Gorodentsev, A.L. (2016). Polynomials and Simple Field Extensions. In: Algebra I. Springer, Cham. https://doi.org/10.1007/978-3-319-45285-2_3

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DOI: https://doi.org/10.1007/978-3-319-45285-2_3
Published: 26 November 2016
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-45284-5
Online ISBN: 978-3-319-45285-2
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