Symmetries

We assume

a^{2} + a v + u = 0

for

u, v \in \bar{Q}

and

u^{2} + u q + p = 0

v^{2} + s v + r = 0

for

p, q, r, s \in Q

. Find some monic polynomial

P \in Q [X]

such that

P (a) = 0

$E$ is the subspace generated by $1, a, u, a u, v, a v, u v, a u v$ ; we assume this is a basis of the eight dimensional space $E$ . Multiplication by $a$ maps these vectors onto the following vectors: $\begin{array}{rcl} 1 & \to & a \\ a & \to & a^{2} = - a v - u \\ u & \to & a u \\ a u & \to & a^{2} u = - a u v - u^{2} = - a u v + q u + p 1 \\ v & \to & a v \\ a v & \to & a^{2} v = - a v^{2} - u v = s a v + r a - u v \\ u v & \to & a u v \\ a u v & \to & a^{2} u v = - a u v^{2} - u^{2} v = a u (s v + r) + (q u + p) v = s a u v + r a u + q u v + p v \end{array}$

Thus the matrix of this map with respect to the basis

1, a, u, a u, v, a v, u v, a u v

is the transpose of

(\begin{array}{cccccccc} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & 0 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ p & 0 & q & 0 & 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & r & 0 & 0 & 0 & s & - 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & r & p & 0 & q & s \end{array})

a

is a root of the characteristic polynomial

P \in Q [X]

of this matrix.

A=matrix(SR,[[0,1,0,0,0,0,0,0],[0,0,-1,0,0,-1,0,0],[0,0,0,1,0,0,0,0],[p,0,q,0,0,0,0,-1],[0,0,0,0,0,1,0,0],[0,r,0,0,0,s,-1,0],[0,0,0,0,0,0,0,1],[0,0,0,r,p,0,q,s]])
A.characteristic_polynomial()

This will give you

\begin{array}{rcl} P & = & X^{8} - 2 s X^{7} + (s^{2} - 2 q + 2 r) X^{6} + (3 q s - 2 r s) X^{5} \\ + (- q s^{2} + q^{2} - 2 q r + r^{2} + 2 p) X^{4} + (- q^{2} s + q r s - 2 p s) X^{3} \\ + (q^{2} r + p s^{2} - 2 p q - 2 p r) X^{2} + p q s X + p^{2} \end{array}

For

p, q, r, s \in Z

we have:

P \in Z [X]

and by Gauß' lemma

P

is irreducible over

Q [X]

iff it's irreducible over

Z [X]

. In order to prove irreducibility of

P

you may utilize Eisenstein's criterion or irreducibility of

P

over some

Z_{n}

for some. In particular for

p = 2, q = 0, r = 3, s = 0

, i.e.

u = \sqrt{2}

v = \sqrt{3}

and

a^{2} + \sqrt{3} u + \sqrt{2} = 0

P = X^{8} + 6 * X^{6} + 13 * X^{4} - 12 * X^{2} + 4

To check irreducibility:

R=PolynomialRing(ZZ,'X')
X=R.gen()
P=X^8 + 6*X^6 + 13*X^4 - 12*X^2 + 4
P.is_irreducible()
P.factor()

which shows that

P

is irreducible.