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Let $F$ be a field of characteristic not equal to $2$.

Assume $L/F$ is a field extension of degree $6$ with no intermediate subfields.

We know by a result due to Joubert (1867), later improved by H. Kraft (2006), that one can find $\alpha$ $\in L$ such that $L = F(\alpha)$ and $min_{\alpha , F}(X)$ = $X^6 + a_4X^4 + a_2X^2 + a_1X^1 + a_0$ where $a_i \in F^{*}$ and $a_1 = a_0 \neq 0$. This result concerns general separable field extensions of degree $6$ in characteristic $\neq 2$. My questions is whether this can be improved under the additional assumption that the Galois group corresponding to the normal closure of $L$ is either $S_6$ or $A_6$.

In particular, I am trying to reduce to the case where $min_{\alpha , F}(X)$ = $X^6 + a_1X^1 + a_0$ or $min_{\alpha , F}(X)$ = $X^6 + a_5X^5 + a_0$.

Any references to the literature outside of that mentioned above is appreciated!

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  • $\begingroup$ First of all, the field extension does not change under the substitution $(a_0,a_1,a_2,a_4)\mapsto (b^6 a_0,b^5a_1,b^4a_2,b^2a_4)$ for $b\in F^\ast$. That means that there are really "$\leq 3$ moduli" of such field extensions. In fact, there are precisely $3$ moduli, so you will never get away with fewer than $3$ undetermined coefficients. $\endgroup$
    – Jason Starr
    Commented Jun 7, 2017 at 21:00
  • $\begingroup$ What can be said along the same lines for extensions of degree 4 with no intermediate subfields? $\endgroup$
    – Gerry Myerson
    Commented Jun 7, 2017 at 23:49
  • $\begingroup$ arxiv.org/pdf/math/0403323.pdf Section 7 answers your question Gerry. $\endgroup$
    – Yasmin
    Commented Jun 8, 2017 at 17:53

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