Inverse Galois problem

The polynomial $x^2-2$ has roots $\sqrt{2}$ and $-\sqrt{2}$. Numbers of the form $a+b\sqrt{2}$, with rational $a$ and $b$, allow two symmetries that preserve arithmetic and every rational number: leave them alone, or flip the sign of $\sqrt{2}$. These symmetries form a two-element group; doing the flip twice returns to the start. Can every finite group arise as all the arithmetic-preserving symmetries of a field obtained by adjoining all roots of a nonconstant rational polynomial? This is a finite Galois extension: finitely many fixed basic ingredients suffice to express every number using rational coefficients, even though the field has infinitely many numbers.[1] [2]

Can every finite group occur as the full group of arithmetic-preserving symmetries of a finite Galois extension of the rational numbers? A field extension retains those rational numbers while adding new ones. One way to obtain a finite Galois extension is to adjoin all roots of a nonconstant polynomial with rational coefficients. The resulting field contains the arithmetic combinations of those roots; its symmetries must fix every rational number.

For $L=\mathbb Q(\sqrt{2})$, every element has the form $a+b\sqrt{2}$ with rational $a,b$. The map $$a+b\sqrt{2}\longmapsto a-b\sqrt{2}$$ preserves sums and products and squares to the identity. It gives one nontrivial symmetry, alongside the identity. The two basis elements $1$ and $\sqrt{2}$ make this a finite extension even though it contains infinitely many numbers.

For any requested finite group $G$, the target is an extension $L/\mathbb Q$ with $$\operatorname{Gal}(L/\mathbb Q)\cong G.$$ The Galois group consists of the field's automorphisms fixing $\mathbb Q$; the isomorphism must preserve their composition law. The universal question asks for every finite group, rather than just groups exhibited by particular polynomials.[1] [2]

There is no general realization theorem over ℚ.[1]

Regular realization over ℚ(t) would suffice by specialization, but is a stronger unresolved target.[1]

Polynomials with rational coefficients can have roots linked by exact arithmetic symmetries. The question asks how freely those symmetries can be prescribed: can every abstract finite group be realized by actual algebraic numbers? An answer would connect the full range of finite symmetry patterns to equations over the rationals.[1]

Galois theory turns rational equations into groups of symmetries of their roots. The inverse problem asks whether this bridge reaches every finite group, so that abstract symmetry structures are all represented by rational arithmetic. Constructing groups over a field of rational functions can use geometric families of equations and then seek rational specializations; the stronger regular realization target would provide a route to realizations over the rationals. This explains the connection between geometric constructions and the specific arithmetic base, without identifying the two targets.[1]