Abstract
Let us identify the Euclidean coordinate plane \(\mathbb{R}^{2}\) with the field \(\mathbb{C}\) in the standard way.
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Notes
- 1.
See Section 3.5.1 of Algebra I.
- 2.
The parametric equation z = a + (b − a) ⋅ t defines the line ℓ a, b as t runs through \(\mathbb{R}\), and defines the circle C a, b as t runs through the unit circle \(\mathop{\mathrm{U}}\nolimits _{1} \subset \mathbb{C}\).
- 3.
See Section 13.3 of Algebra I.
- 4.
See Theorem 13.7 on p. 313.
- 5.
Obtained from the relation cos(3φ) = 4cosφ − 3cos2φ for φ = π∕9.
- 6.
Recall that the cyclotomic polynomial \(\Phi _{p}(x)\) is irreducible for prime \(p \in \mathbb{N}\) by Eisenstein’s criterion; see Example 5.9 of Algebra I.
- 7.
See Example 13.6 on p. 313.
- 8.
- 9.
Compare with Section 3.6.3 of Algebra I.
- 10.
See the discussion after formula (3.22) of Algebra I.
- 11.
See Sect. 5.4.2 on p. 111.
- 12.
In addition to the already cited Sect. 5.4.2, see Exercise 5.15 on p. 111.
- 13.
See Section 13.3.1 of Algebra I.
- 14.
Note that a 1, a 2, …, a n are polynomials in t 1, t 2, …, t n by Viète’s theorem.
- 15.
See Problem 12.3 on p. 294 and Example 13.1 on p. 298.
- 16.
See Section 3.6.3 of Algebra I and compare this problem with Problems 3.38 and 9.7 from Algebra I.
- 17.
That is, the integral closure of \(\mathbb{Z}\) in \(\mathbb{K}\).
- 18.
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). Examples of Galois Groups. In: Algebra II. Springer, Cham. https://doi.org/10.1007/978-3-319-50853-5_14
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