Let $f \in K[X]$ a polynomial of degree $m$ and let $\theta_1, \ldots, \theta_n$ it's roots (in an algebraic closure $\overline{K}$ of $K$), let $N := K(\theta_1, \ldots, \theta_m)$ be the splitting field of $f$ over $K$ and let $$ R := \{ r(X_1, \ldots, X_m) \in K[X_1, \ldots, X_m] : r(\theta_1, \ldots, \theta_m) = 0 \} $$ the set of $K$-rational relations between $\theta_1, \ldots, \theta_2$, then $$ \operatorname{Gal}(f) := \{ \sigma \in S_m : r(\theta_{\sigma(1)}, \ldots, \theta_{\sigma(m)}) = 0, r(X_1, \ldots, X_m) \in R \} $$ a subgroup of the symmetric group $S_m$, which is called the Galois group of $f$.

Intuitively, the Galois group consists of those permutations that respect all algebraic relations. So if there are no relations (i.e. $R = \emptyset$) then the Galois group is the full symmetric group (i.e. $Gal(f) = S_m$).

Now my question, how can I construct a polynomial such that no relations between the root hold, for example $x^3 - 2$ is such, but how to construct it in general? That his how to find a polynomial such that it's Galoisgroup is $S_m$?

  • $\begingroup$ Is $K$ given, or can we choose $K$ to make the question easy? $\endgroup$ – Mark Bennet Jul 25 '13 at 21:13
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    $\begingroup$ If $m$ is prime, the Galois group of an irreducible polynomial is transitive and therefore contains a cycle of order $m$ which is necessarily an $m$-cycle (e.g. (123..m)). If the polynomial has a complex root, then complex conjugation acts nontrivially giving a 2-cycle. Then as any 2-cycle and any $m$-cycle must generate all of $S_m$, the Galois group is $S_m$. $\endgroup$ – RghtHndSd Jul 25 '13 at 21:17
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    $\begingroup$ @rghthndsd You seem to be assuming we are looking at polynomials over $K= \mathbb Q$ - but it is not clear we have characteristic $0$. If $K=\mathbb R$ we can get only $S_2$ and if $K=\mathbb C$ we don't even get that. $\endgroup$ – Mark Bennet Jul 25 '13 at 21:54
  • $\begingroup$ I just had $\mathbb Q$ in mind, but for general $K$ might be more interesting. $\endgroup$ – StefanH Jul 25 '13 at 22:23

There is a construction of a polynomial with $S_n$ in:

S. Lang, Algebra, 3rd ed., 2002, Graduate Texts in Mathematics, 211, Springer

Chapt. VI "Galois Theory", Example 4, p.272.


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