The Inverse Galois Problem, Hilbertian Fields, and Hilbert’s Irreducibility Theorem
نویسنده
چکیده
In the study of Galois theory, after computing a few Galois groups of a given field, it is very natural to ask the question of whether or not every finite group can appear as a Galois group for that particular field. This question was first studied in depth by David Hilbert, and since then it has become known as the Inverse Galois Problem. It is usually posed as which groups appear as Galois extensions over Q specifically, and there have been a number of celebrated results over the years pertaining to this question, perhaps most notably being Shafarevich’s theorem that every solvable group has such a realization over Q. This paper, however, will focus around the classical result known today as Hilbert’s Irreducibility Theorem, which is a useful tool in Inverse Galois Theory. A large number of realizations of groups as Galois groups over Q can easily be found using basic types of extensions. For instance, using cyclotomic field extensions and the theorem of finitely generated abelian groups, one finds that every finite abelian group can be realized. We also have that if K and K ′ are two Galois extensions of a field F , where K∩K ′ = F , then KK ′ is also a Galois extension, with Galois group Gal(K/F ) × Gal(K ′/F ), so we can often realize the direct product of two realizable groups. Given a Galois extension K with Galois group G and normal subgroup H, we have by the fundamental theorem of Galois that G/H can be realized over the same field as well. A very important group we may be curious about realizing as a Galois group over Q is Sn. By looking at the general polynomial f(x) = (x − x1) · · · (x − xn), where x1, . . . , xn are indeterminate and s1, . . . , sn are the elementary symmetric functions of x1, . . . , xn, we can find that the function field Q(x1, . . . , xn) is Galois over Q(s1, . . . , sn), with Galois group Sn. We would like, however, to find Sn realized as a Galois extension over Q instead. In general, when we have a Galois group realized as an extension of a field of rational functions over a field F , we
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