Defining the p-Frobenius for $GF(p^{16})$ explicit

Question

Basics

First let me tell you some background. Consider $p\equiv 13 \bmod 32$, therefore, $p\equiv 5\bmod 8$. That means, 2 is a none-quadratic residue in $GF(p)$. Therefore, we are able to build up $\newcommand\F{\mathbb F} \F_{p^{16}}:=GF(p^{16})\cong \F_p[x]/(x^{16}-2)$.

That means we have the identity $x^{16}=2$ and $2^{\frac{p-1}{2}}\equiv -1\bmod p$ (Quadratic reciprocity, 2nd supplement). Now we have

For any $A\in\F_{p^{16}}$ there are $a_0,...,a_{15}\in\F_p$ such that $A=\sum_{i=0}^{15} a_ix^i$.

Now consider the p-Frobenius:

$\pi_p(a)=A^p= \sum_{i=0}^{15} a_i \left(x^i\right)^p$, since $a_i^p \equiv a_i \bmod p$ from Fermats little theorem.

That means, all we have to do is to produce valid output for $x^{ip}$ for any $1\leq i \leq 15$.

Let me start with i=15 (the easy one..):

$\left( x^{15} \right)^p = \left( x^{16} \right)^{\frac{p-1}{2}}x^2=-x^2$ Edit: {Oh, I did wrong. This is not equal. I cannot take out the $x^2$ part in this way...}

Question

Now I'm out of ideas to deal with the rest where the Question starts. This might be solved by computer-power, but I don't know how to. Hints for some steps would be great. I do not need a complete solution, I would like to master this for my own.

Attemption

My aim is always: Reaching an expression $(x^{16})^{\frac{p-1}{2}}$ or $(x^{16})^{\frac{p^2-1}{8}}$ which will be equivalent to -1.

Sage attemption with the lates suggested prime for KSS16 in pairings:

Defining $R16 := GF(p^{16})$

#p=615623382030675150502066218751443438064107566348210118507940234835256709422634902533028653925239565581

p=0x465d6f16f520984b92d62d59cf104144153639b6d4c7d8047c9095fa1068d6fda7b640c1c46ac30472d0dL
R = GF(p) 
_.<x> = PolynomialRing(R)
R16.<x> = R.extension(x^16 - 2, 'x')

calling

(x^15)^p

output

599280983495543796770673976542410445676498658176691546033897360530150567095232487071355883922261646714 * x^3

which is different from the one I figured out.

Edit newest attemption

First we have: $p\equiv 13\bmod 16$ then $\left( x^{15} \right)^p = \left( x^{15\cdot 16} \right)^{\frac{p-13}{16}}x^{13} = 2^{15\cdot \frac{p-13}{16}}x^{13}$ which won't lead to the sage-expression.

Answer 1

You didn't prove that $x^{16}-2 \in \mathbb{F}_p[x]$ is irreducible whenever $p\equiv 13 \bmod 32$.

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If $\alpha^{16} = 2$ then ` $$x^{16}-2 = \prod_{k=1}^{16} (x-\zeta_{16}^k \alpha)$$

and there is a primitive $16$-th root of unity such that $\pi_p(\alpha) = \zeta_{16} \alpha$, $\pi_p^2(\alpha) = \pi_p(\zeta_{16})\zeta_{16} \alpha = \zeta_{16}^{p+1}\alpha$, $\pi_p^m(\alpha)=\zeta_{16}^{(p^m-1)/(p-1)}\alpha$ and the minimal polynomial of $\alpha$ is $$f(x) = \prod_{m=1}^N (x-\pi_p^m(\alpha)) = \prod_{m=1}^N (x-\zeta_{16}^{(p^m-1)/(p-1)} \alpha) \quad \in \mathbb{F}_p[x]$$ where $N$ is the least integer such that $16 \ | \ \frac{p^N-1}{p-1}$. If $p = 1+ 2^l (2a+1)$ it means $2^{l+4} \ | \ p^N-1$ so that $N$ is the order of $p \bmod 2^{l+4}$.

If $p \equiv 13 \bmod 64$ or $p \equiv 13+32 \bmod 64$ then $l = 2$ and $ord(13 \bmod 2^{l+4}) = ord(13 +32\bmod 2^{l+4}) = 16$ thus yes $x^{16}-2 \in \mathbb{F}_p[x]$ is irreducible.