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2017 | OriginalPaper | Chapter

13. Algebraic Field Extensions

Author : Alexey L. Gorodentsev

Published in: Algebra II

Publisher: Springer International Publishing

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Abstract

Recall that a field extension \(\mathbb{k} \subset \mathbb{F}\) is said to be finite of degree d if \(\mathbb{F}\) has dimension d < ∞ as a vector space over \(\mathbb{k}\). We write \(\deg \mathbb{F}/\mathbb{k} = d\) in this case.

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previous chapter Algebraic Manifolds

next chapter Examples of Galois Groups

Compare with Sect. 3.4 of Algebra I.

Recall that the discriminant of a monic polynomial f(x) = ∏(x −ϑ _i) is the product \(D(\,f)\stackrel{\mathrm{def}}{=}\prod _{i<j}(\vartheta _{i} -\vartheta _{j})^{2}\) expressed as a polynomial in the coefficients of f.

See Sect. 3.3.3 of Algebra I.

See Sect. 2.4.3 of Algebra I.

The same argument shows that every finite field, considered as an algebra over a subfield, is generated by one element. Although the separability assumption is not used explicitly in this case, we know from Example 3.4 of Algebra I that all finite fields are separable over their prime subfields.

Recall that the degree \(\deg _{\mathbb{k}}a\) of an algebraic element a over a field \(\mathbb{k}\) is the degree of the minimal polynomial μ _a of a over \(\mathbb{k}\).

See Lemma 1.3 of Algebra I.

Automatically distinct.

See Lemma 1.1 of Algebra I.

That is, there exists an isomorphism between them acting on \(\mathbb{k}\) identically.

Such an extension exists by Theorem 3.1 of Algebra I.

With respect to inclusions.

Note that k may be less than m, because the adjunction of a root may cause the appearance of several more roots.

Or equivalently, algebraic.

See Problem 1.20 of Algebra I.

See Sect. 1.4.2 of Algebra I.

See Exercise 1.16 in Sect. 1.4.2 of Algebra I.

It is isomorphic to \(\mathbb{Q}\) for \(\mathop{\mathrm{char}}\nolimits (\mathbb{k}) = 0\), and to \(\mathbb{F}_{p} = \mathbb{Z}/(p)\) for \(\mathop{\mathrm{char}}\nolimits (\mathbb{k}) = p > 0\); see Sect. 2.8.1 of Algebra I.

The proof of Theorem 13.6 is based on Corollary 13.1 on p. 13.1, a direct corollary of Theorem 13.2.

See Exercise 11.14 on p. 256.

See Example 12.4 of Algebra I.

Recall that every subfield \(\mathbb{F} \subset \mathbb{k}(t)\) strictly larger than \(\mathbb{k}\) is isomorphic to \(\mathbb{k}(\,f)\) for some \(f \in \mathbb{k}(t)\) by Lüroth’s theorem; see Theorem 10.4 on p. 239.

Note that even if these eigenvectors are not defined over \(\mathbb{k}\), the G-invariant polynomial with roots at these points has to lie in \(\mathbb{k}[t_{0},t_{1}]\).

The rational functions of t = t ₀∕t ₁ are exactly those polynomials.

Instead of Corollary 13.3, we could have used Theorem 13.6, which says that the extension \(\mathbb{K}^{H} \subset \mathbb{K}\) is a Galois extension with Galois group H for every subgroup H ⊂ G.

Recall that we write D _n for the group of the regular n-gon, which has order 2n; see Example 12.4 of Algebra I.

Note that this model of \(\mathbb{P}_{1}(\mathbb{C})\) differs slightly from that used in Example 11.1 of Algebra I, where the sphere of diameter 1 was projected from the north and south poles onto the tangent planes drawn through the opposite poles.

That is, bijections \(M\stackrel{\sim }{\rightarrow }M\) induced by the orientation-preserving linear isometries \(\mathbb{R}^{3}\stackrel{\sim }{\rightarrow }\mathbb{R}^{3}\); see Sect. 12.3 of Algebra I.

In this case, the extension \(\mathbb{K} \supset \mathbb{k}\) is called purely inseparable.

[DK]

Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux, Russian Math. Surveys 60:2 (2005), 269–334.MathSciNetCrossRef

[Fu]

Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.MATH

[FH]

Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997.MATH

[Mo]

Morris, S. A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977.

Title: Algebraic Field Extensions
Author: Alexey L. Gorodentsev
Publisher: Springer International Publishing
Book: Algebra II
Print ISBN: 978-3-319-50852-8

Electronic ISBN: 978-3-319-50853-5

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-3-319-50853-5_13

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