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Algebra II
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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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.
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.
Recall that we write
D
n for the group of the regular
n-gon, which has order 2
n; 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]
Zurück zum Zitat Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux, Russian Math. Surveys 60:2 (2005), 269–334. MathSciNetCrossRef Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux,
Russian Math. Surveys 60:2 (2005), 269–334.
MathSciNetCrossRef
[Fu]
Zurück zum Zitat Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997. MATH Fulton, W.:
Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.
MATH
[FH]
Zurück zum Zitat Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997. MATH Fulton, W., Harris, J.:
Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997.
MATH
[Mo]
Zurück zum Zitat Morris, S. A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977. Morris, S. A.:
Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977.
- Titel
- Algebraic Field Extensions
- DOI
- https://doi.org/10.1007/978-3-319-50853-5_13
- Autor:
-
Alexey L. Gorodentsev
- Verlag
- Springer International Publishing
- Sequenznummer
- 13
- Kapitelnummer
- Chapter 13