There are many cubic curves that have no such point, for example when " K " is the rational number field.
2.
The tensor product of End ( " A " ) with the rational number field "'Q "', should contain a commutative subring of order in an imaginary quadratic field.
3.
Taking " n " = 3, there are no degree 3 Kummer extensions of the rational number field "'Q "', since for three cube roots of 1 complex numbers are required.
4.
The abelian group of divisor classes up to algebraic equivalence is now called the N�ron-Severi group; it is known to be a finitely-generated abelian group, and the result is about its tensor product with the rational number field.
5.
This tells us that each root of has-adic valuation and hence that is irreducible over the-adic field ( since, for instance, no product of any proper subset of the roots has integer valuation ); and " a fortiori " over the rational number field.
6.
The rational number field "'Q "'is Hilbertian, because Hilbert's irreducibility theorem has as a corollary that the projective line over "'Q "'is Hilbertian : indeed, any algebraic number field is Hilbertian, again by the Hilbert irreducibility theorem.
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