Exercise 12.2.13

Question

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers 
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\be} {\begin{enumerate}}

ewcommand{\ee} {\end{enumerate}}

ewcommand{\deb}{\begin{eqnarray*}}

ewcommand{\fin}{\end{eqnarray*}}

ewcommand{\ssi} {si et seulement si }

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{\U}{\mathbb{U}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\im}{\,\mathrm{Im}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{\Gal}{\mathrm{Gal}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

This exercise will show that not all choices of the $t_i$ in (12.21) give Galois resolvents. As in Example 12.2.1, $f = (x^2-2)(x^2-3)$ has roots $\sqrt{2},-\sqrt{2},\sqrt{3}$, and $-\sqrt{3}$. This time we will use $(t_1,t_2,t_3,t_4) = (0,1,2,3)$. Show that (12.21) gives the polynomial
\begin{align*}
s(y) &= 1679616 - 45722880y^2 + 445417056y^4-1935550800y^6\
&+4169468065y^8-4504515400y^{10}+2268233020y^{12}-432170200y^{14}\
&+36781990y^{16}-1483000y^{18}+29596y^{20} -280y^{22} + y^{24}\
&=(81-90y^2+y^4)^2 (16-40y^2+y^4)^2(1-10y^2+y^4)^2.
\end{align*}
This does not have distinct roots, so that $s(y)$ is not a Galois resolvent.

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers 
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\be} {\begin{enumerate}}

ewcommand{\ee} {\end{enumerate}}

ewcommand{\deb}{\begin{eqnarray*}}

ewcommand{\fin}{\end{eqnarray*}}

ewcommand{\ssi} {si et seulement si }

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{\U}{\mathbb{U}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\im}{\,\mathrm{Im}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{\Gal}{\mathrm{Gal}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

\medskip

Note. The results in Example 12.2.1 are false for $(t_1,t_2,t_3,t_4) = (0,1,2,4)$. The first given factor of $s(y)$ is $900 -132 y^2 + y^4$, which  has root $\sqrt{66 + 24 \sqrt{6}} = 3\sqrt{2} + 4 \sqrt{3}$ and this root can't be written $t_{\sigma(1)}\sqrt{2} + t_{\sigma(2)}(-\sqrt{2}) + t_{\sigma(3)}\sqrt{3} + t_{\sigma(4)}(-\sqrt{3})$ for any permutation $\sigma \in S_4$. Idem for the second factor $25 - 118 y^2 + y^4$.

The following Sage instructions gives the right answer : 
\begin{verbatim}
t1,t2,t3,t4 = 0,1,2,4
var('x1,x2,x3,x4')
V = t1*x1 + t2*x2 + t3*x3 + t4*x4
from itertools import permutations
R.<y> = ZZ[]
t = 1
for perm in permutations([x1,x2,x3,x4]):
    t = t * (y - V.subs(x1 = perm[0], x2 = perm[1], x3 = perm[2], x4 = perm[3]))
s0= t.subs(x1 = sqrt(2),x2 = -sqrt(2), x3 = sqrt(3),x4 = -sqrt(3))
s = R(s0.expand())
s
\end{verbatim}
\begin{align*}
s(y) = & \ y^{24} - 350y^{22} + 52395y^{20} - 4390200y^{18} + 226512195y^{16} -7470312150y^{14}\
 &+ 158533048725y^{12} - 2128033120500y^{10}+17319964832940y^{8} - 79514980673600y^{6}\
  &+ 185487963684016y^{4} -182187606350400y^{2} + 57817774440000
\end{align*}

and
\begin{verbatim}
s.factor()
\end{verbatim}
gives the Galois resolvent $s(y)$:
$$(y^{4} - 100y^{2} + 2116) \cdot (y^{4} - 70y^{2} + 361) \cdot (y^{4} -70y^{2} + 841) \cdot (y^{4} - 60y^{2} + 36) \cdot (y^{4} - 28y^{2} +100) \cdot (y^{4} - 22y^{2} + 25)$$

The minimal polynomial of $V = -\sqrt{2} - 2 \sqrt{3}$ is the factor $h = y^{4} - 28y^{2} +100$.
\begin{proof}

The same instructions with $t_1,t_2,t_3,t_4 = 0,1,2,3$ give
\begin{align*}
s(y) =&\  y^{24} - 200y^{22} + 16620y^{20} - 743400y^{18} + 19430070y^{16} -
302989800y^{14}\
& + 2777491500y^{12} - 14100111000y^{10} +
34064189265y^{8} - 25798725200y^{6}\
& + 7753861216y^{4} - 910060800y^{2} +
36000000\
= &\ (y^{4} - 58y^{2} + 625) \cdot (y^{4} - 42y^{2} + 225) \cdot (y^{4} -
40y^{2} + 16)^{2} \cdot (y^{4} - 10y^{2} + 1)^{2}
\end{align*}
This does not have distinct roots, so that $s(y)$ is not a Galois resolvent.

(But the result is not the same as in the statement.)
\end{proof}

Exercise 12.2.13

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Exercise 12.2.13

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