The question deals with the finite field $\mathrm{GF}(p^n)$. It wants to study/attack a variant of DSA in (the multiplicative subgroup of) that field, rather than $\mathrm{GF}(p)$. Whatever that DSA variant is, it can be attacked by solving the Discrete Logarithm Problem in $\mathrm{GF}(p^n)$.
The multiplicative subgroup of $\mathrm{GF}(p^n)$ has order $p^n-1$, and the order of any element divides that. If the order of $a$ has no large prime factor, then the Pohlig-Hellman method allows reduction to easier subproblems, that a general DLP algorithm like Baby-Step Giant Step or Pollard's rho can solve. DSA blocks such attacks by working in a multiplicative subgroup of large prime order $q$ (e.g. at least 256-bit). That also keeps signature size to the minimum (twice the bitsize of $q$).
The parameters envisioned are $p\approx10^{10}$ and $n=17$, thus $p^n$ about 565-bit, making such subgroup plausible. If we want $p$ just below $2^{32}$ and $x^{17}-2$ irreducible modulo $p$ (which simplifies computations), we can use the 32-bit $p=\mathtt{ffe083f1}_\text{h}$, with $p^{17}-1$ having as highest prime factor the 262-bit $q=\mathtt{2a92bc16243adf3264304adf2adc292c32e64b569a5abfbd7be71d7a1816e3d8c1}_\text{h}$. To obtain an element of order $q$, we can select almost any arbitrary polynomial $u$; find the order $r$ of $u$ (we divide $p^{17}-1$ by primes in its factorization while raising $u$ to that power yields $1$); and use $g=u^{r/q}$ as the generator of a subgroup of order $q$.
Building a DSA analog from that and SHA-512 is easy. The only difficulty is when the original makes an exponentiation modulo $p$ followed by a reduction modulo $q$. Among many, one option to parallel this is to make the exponentiation modulo $x^{17}-2\pmod p$, then evaluate that polynomial in $\Bbb Z$ at the point $x=2^{32}$ (that is concatenate the coefficients expressed as 32-bit integers), then reduce modulo $q$.
That will make a working DSA, secure against all the aforementioned algorithms thanks to the large $q$. But that's nevertheless insecure, for other DLP methods are applicable to the case at hand. There's a summary of recent results in Razvan Barbulescu and Cécile Pierrot's The Multiple Number Field Sieve for Medium and High Characteristic Finite Fields, in LMS Journal of Computation and Mathematics, 2014.
Tentatively: the algorithm of choice to attack that DSA analog could be Antoine Joux, Reynald Lercier, Nigel P. Smart, and Frederik Vercautere's The number field sieve in the medium prime case, in proceedings of Crypto 2006.
The musing with a small example at the end of the question attempts to work in $\mathrm{GF}(17^7)$, but uses a polynomial $m_1=x^7-2$ that is not irreducible modulo $p=17=m_2$, since$$(x+8)(x^6+9x^5+13x^4+15x^3+16x^2+8x+4)\\\quad\equiv x^7-2\pmod{17}$$
It follows that $(x+8)$ is not invertible, and we get a ring rather than a field.
If we change to $p=29$ that makes $x^7-2$ irreducible. I assume this in the following.
The multiplicative subgroup of $\mathrm{GF}(p^n)$ has order $p^n-1$, here $29^7-1$. This factors as $2^2\cdot7^2\cdot88009573$. The order of $a$ is bound to be a divisor of that. By computing $a^{(29^7-1)/2}=-1\ne1$, $a^{(29^7-1)/7}=-5\ne1$, and $a^{(29^7-1)/88009573}\ne1$, we get that $a$ is a generator of the full multiplicative group. $b\ne0$, therefore $a^e=b$ has a solution.
