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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Notes
- 1.
Compare with Sect. 3.4 of Algebra I.
- 2.
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.
- 3.
See Sect. 3.3.3 of Algebra I.
- 4.
See Sect. 2.4.3 of Algebra I.
- 5.
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.
- 6.
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 \(\mathbb{k}\).
- 7.
See Lemma 1.3 of Algebra I.
- 8.
Automatically distinct.
- 9.
See Lemma 1.1 of Algebra I.
- 10.
That is, there exists an isomorphism between them acting on \(\mathbb{k}\) identically.
- 11.
Such an extension exists by Theorem 3.1 of Algebra I.
- 12.
With respect to inclusions.
- 13.
Note that k may be less than m, because the adjunction of a root may cause the appearance of several more roots.
- 14.
Or equivalently, algebraic.
- 15.
See Problem 1.20 of Algebra I.
- 16.
See Sect. 1.4.2 of Algebra I.
- 17.
See Exercise 1.16 in Sect. 1.4.2 of Algebra I.
- 18.
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.
- 19.
- 20.
See Exercise 11.14 on p. 256.
- 21.
See Example 12.4 of Algebra I.
- 22.
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.
- 23.
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 \(\mathbb{k}[t_{0},t_{1}]\).
- 24.
The rational functions of t = t 0∕t 1 are exactly those polynomials.
- 25.
- 26.
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.
- 27.
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.
- 28.
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.
- 29.
In this case, the extension \(\mathbb{K} \supset \mathbb{k}\) is called purely inseparable.
References
Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux, Russian Math. Surveys 60:2 (2005), 269–334.
Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.
Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997.
Morris, S. A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977.
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Gorodentsev, A.L. (2017). Algebraic Field Extensions. In: Algebra II. Springer, Cham. https://doi.org/10.1007/978-3-319-50853-5_13
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