It's easier not to work with the field of all algebraic numbers, but with finite extensions of the rationals, e . g . the number field generated by some given algebraic number u : K = Q ( u ).
32.
If " N " is a Galois extension of a Hilbertian field, then although " N " need not be Hilbertian itself, Weisseauer's results asserts that any proper finite extension of " N " is Hilbertian.
33.
Analogously, the group of p-adic numbers \ Q _ p is isomorphic to its dual . ( In fact, any finite extension of \ Q _ p is also self-dual . ) It follows that the adeles are self-dual.
34.
If " L " is a finite extension of " K " that is separable ( for example, this is automatically satisfied if " K " is finite or has characteristic zero ) then the following property is also equivalent:
35.
In case " K " and " L " are finite extensions of N, the situation is particularly simple since the tensor product is of finite dimension as an " N "-algebra ( and thus an Artinian ring ).
36.
The p-adic fields or any finite extension of them are characteristic zero fields, much applied in number theory, that are constructed from rings of characteristic " p " " k ", as " k " ?! ".
37.
An extension " E " / " F " is also sometimes said to be simply "'finite "'if it is a finite extension; this should not be confused with the fields themselves being finite fields ( fields with finitely many elements ).
38.
In a sense, a finitely generated extension is a transcendental generalization of a finite extension since, if the generators in " A " are all algebraic, then " F " ( " A " ) is a finite extension of " F ".
39.
In a sense, a finitely generated extension is a transcendental generalization of a finite extension since, if the generators in " A " are all algebraic, then " F " ( " A " ) is a finite extension of " F ".
40.
Let be a number field ( i . e ., a finite extension of \ mathbb Q, the set of rational numbers ), in other words, K = \ mathbb { Q } ( \ theta ) for some algebraic number \ theta \ in \ mathbb { C } by the primitive element theorem.
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