\geq m(hf',\mu) = m(f',\mu')$. Since $\mu$ and $\mu'$ are conjugates over ${\bar K}$, we see that this implies that $\nu(f',\mu) \geq \nu(f',\mu')$. This proves that $\mu$ is an instability 1-PS of $f'$ and hence completes the proof of Claim 2.

Now since $f$ and $f'$ are translates of each other by an element in $K^{1/p^{t_1}}$ and an instability 1-PS of $f'$ is defined over $K$, we see that an instability 1-PS and hence the instability parabolic of $f$ is defined over $K^{1/p^{t_1}}$.

Let $\rho : G \rightarrow S^N(V)$ be the representation as in lemma [\[Sl(2)\]]{}. If $N > p$, the instability parabolic of any non-semistable vector in $S^N(V)$ is rational.

Obvious.

In general, for an arbitrary representation of $Sl(V)$, the method does not seem to work. This is because it is in general impossible to determine all the non-semistable points in the representing space. Hence we have adopt a more indirect way of bounding the field of definition of the instability 1-PS which does not use the knowledge of all the non-semistable vectors. We begin with a lemma which will be a crutial step in the bounding of the field of definition of the instability 1-PS :

[\[Noether normalization\]]{} Let 