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Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/113081
Realising $S_n$ and $A_n$ as Galois groups Over $\mathbb{Q}$ : an introduction to the inverse Galois problem
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Given a field $k$ and a finite group $G$, is there a Galois field extension $K|k$ such that its Galois group is isomorphic to $G$? Such an innocent question and yet it remains unsolved: this is what is known as the Inverse Galois Problem. In the present Bachelor thesis we show that this question has a positive answer if
the field is $\mathbb{Q}$ and the group is either $S_n$ or $A_n$, following the strategy devised by David Hilbert in his paper Über die Irreduzibilität ganzer rationaler Funktionen mit ganzzahligen Koeffizienten (1892). We start with two basic examples and an exposition of relevant results from algebraic number theory, and then move on to proving Hilbert’s Irreducibility Theorem. As a consequence, we prove that the symmetric group $S_n$ and the alternating group $A_n$ are realisable as Galois groups over the field of rational numbers $\mathbb{Q}$.
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Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2017, Director: Núria Vila i Oliva
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MARTÍNEZ I SELLARÈS, Mireia. Realising $S_n$ and $A_n$ as Galois groups Over $\mathbb{Q}$ : an introduction to the inverse Galois problem. [consulted: 11 of October of 2026]. Available at: https://hdl.handle.net/2445/113081