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Here is a belated answer (I came across this question only today) which doesn't require anything more than Kummer theory --- or Artin-Schreier theory, if you want to allow $K$ to be a finite extension of ${\mathbf F}_l((t))$. I will confine myself to the more interesting case of degree-$l$ extensions (where $l$ is the residual characteristic).

If $K$ contains a primitive $l$-th root of $1$, then there is a natural bijection between the set of degree-$l$ cyclic extensions of $K$ and the set of ${\mathbf F}_l$-lines in $K^\times/K^{\times l}$ (Kummer theory). The structure of this filtered ${\mathbf F}_l$-space is completely known; see for example Section V of arXiv:0711.3878 (where your $l$ is called $p$).

If $K$ does not contain a primitive $l$-th root $\zeta$ of $1$, then put $K'=K(\zeta)$, $\Delta={\rm Gal}(K'|K)$ and $\omega:\Delta\to{\mathbf F}_l^\times$ the cyclotomic character giving the action of $G$ on the $l$-th roots of $1$. Then there is a natural bijection between the set of degree-$l$ cyclic extensions of $K$ and ${\mathbf F}_l$-lines in the $\omega$-eigenspace for the action of $\Delta$ on $K^{\prime\times}/K^{\prime\times l}$. The structure of this filtered ${\mathbf F}_l[\Delta]$-module can be completely determined; see for example arXiv:0912.2829.

If you are interested more generally in all degree-$l$ (separable) extensions of $K$ (and not just the cyclic ones), then something similar can be done. Put $L=K(\root{l-1}\of{K^\times})$ and $G={\rm Gal}(L|K)$. There is a natural bijection between the set of (isomorphism classes of) degree-$l$ (separable) extensions of $K$ and the set of $G$-stable lines in the ${\mathbf F}_l$-space $L^\times/L^{\times l}$. Again, the structure of this filtered ${\mathbf F}_l[G]$-module is completely known: see for example arXiv:1005.2016.

Finally, if you want to replace your $K$ by a finite extension of ${\mathbf F}_l((t))$, there are similar results using Artin-Schreier theory instead of Kummer theory. See for example arXiv:0909.2541 for degree-$l$ cyclic extensions and arXiv:1005.2016 for degree-$l$ separable extensions. They allow you to count the number of extensions with bounded ramification, of which there are only finitely many.

Here is a belated answer (I came across this question only today) which doesn't require anything more than Kummer theory --- or Artin-Schreier theory, if you want to allow $K$ to be a finite extension of ${\mathbf F}_l((t))$. I will confine myself to the more interesting case of degree-$l$ extensions (where $l$ is the residual characteristic).

If $K$ contains a primitive $l$-th root of $1$, then there is a natural bijection between the set of degree-$l$ cyclic extensions of $K$ and the set of ${\mathbf F}_l$-lines in $K^\times/K^{\times l}$ (Kummer theory). The structure of this filtered ${\mathbf F}_l$-space is completely known; see for example Section V of arXiv:0711.3878 (where your $l$ is called $p$).

If $K$ does not contain a primitive $l$-th root $\zeta$ of $1$, then put $K'=K(\zeta)$, $\Delta={\rm Gal}(K'|K)$ and $\omega:\Delta\to{\mathbf F}_l^\times$ the cyclotomic character giving the action of $G$ on the $l$-th roots of $1$. Then there is a natural bijection between the set of degree-$l$ cyclic extensions of $K$ and ${\mathbf F}_l$-lines in the $\omega$-eigenspace for the action of $\Delta$ on $K^{\prime\times}/K^{\prime\times l}$. The structure of this filtered ${\mathbf F}_l[\Delta]$-module can be completely determined; see for example arXiv:0912.2829.

If you are interested more generally in all degree-$l$ (separable) extensions of $K$ (and not just the cyclic ones), then something similar can be done. Put $L=K(\root{l-1}\of{K^\times})$ and $G={\rm Gal}(L|K)$. There is a natural bijection between the set of (isomorphism classes of) degree-$l$ (separable) extensions of $K$ and the set of $G$-stable lines in the ${\mathbf F}_l$-space $L^\times/L^{\times l}$. Again, the structure of this filtered ${\mathbf F}_l[G]$-module is completely known: see for example arXiv:1005.2016.

Finally, if you allow $K$ to be a finite extension of ${\mathbf F}_l((t))$, there are similar results using Artin-Schreier theory instead of Kummer theory. See for example arXiv:0909.2541 for degree-$l$ cyclic extensions (which correspond to ${\mathbf F}_l$-lines in $K^+/\wp(K^+)$, where $\wp(x)=x^l-x$) and arXiv:1005.2016 for degree-$l$ separable extensions, which correspond to $G$-stable ${\mathbf F}_l$-lines in $L^+/\wp(L^+)$, where $L$ is still $K(\root{l-1}\of{K^\times})$ and $G={\rm Gal}(L|K)$. The filtered ${\mathbf F}_l$-space (resp. ${\mathbf F}_l[G]$-module) $K^+/\wp(K^+)$ (resp. $L^+/\wp(L^+)$) has been completely determined therein. These results allow you in particular to count the number of extensions with bounded ramification, of which there are only finitely many.

Here is a belated answer (I came across this question only today) which doesn't require anything more than Kummer theory --- or Artin-Schreier theory, if you want to allow $K$ to be a finite extension of ${\mathbf F}_l((t))$. I will confine myself to the more interesting case of degree-$l$ extensions (where $l$ is the residual characteristic).

If $K$ contains a primitive $l$-th root of $1$, then there is a natural bijection between the set of degree-$l$ cyclic extensions of $K$ and the set of ${\mathbf F}_l$-lines in $K^\times/K^{\times l}$ (Kummer theory). The structure of this filtered ${\mathbf F}_l$-space is completely known; see for example Section V of arXiv:0711.3878 (where your $l$ is called $p$).

If $K$ does not contain a primitive $l$-th root $\zeta$ of $1$, then put $K'=K(\zeta)$, $\Delta={\rm Gal}(K'|K)$ and $\omega:\Delta\to{\mathbf F}_l^\times$ the cyclotomic character giving the action of $G$ on the $l$-th roots of $1$. Then there is a natural bijection between the set of degree-$l$ cyclic extensions of $K$ and ${\mathbf F}_l$-lines in the $\omega$-eigenspace for the action of $\Delta$ on $K^{\prime\times}/K^{\prime\times l}$. The structure of this filtered ${\mathbf F}_l[\Delta]$-module can be completely determined; see for example arXiv:0912.2829.

If you are interested more generally in all degree-$l$ (separable) extensions of $K$ (and not just the cyclic ones), then something similar can be done. Put $L=K(\root{l-1}\of{K^\times})$ and $G={\rm Gal}(L|K)$. There is a natural bijection between the set of (isomorphism classes of) degree-$l$ (separable) extensions of $K$ and the set of $G$-stable lines in the ${\mathbf F}_l$-space $L^\times/L^{\times l}$. Again, the structure of this filtered ${\mathbf F}_l[G]$-module is completely known: see for example arXiv:1005.2016.

Finally, if you want to replace your $K$ by a finite extension of ${\mathbf F}_l((t))$, there are similar results using Artin-Schreier theory instead of Kummer theory. See for example arXiv:0909.2541 for degree-$l$ cyclic extensions and arXiv:1005.2016 for degree-$l$ separable extensions.

Here is a belated answer (I came across this question only today) which doesn't require anything more than Kummer theory --- or Artin-Schreier theory, if you want to allow $K$ to be a finite extension of ${\mathbf F}_l((t))$. I will confine myself to the more interesting case of degree-$l$ extensions (where $l$ is the residual characteristic).

If $K$ contains a primitive $l$-th root of $1$, then there is a natural bijection between the set of degree-$l$ cyclic extensions of $K$ and the set of ${\mathbf F}_l$-lines in $K^\times/K^{\times l}$ (Kummer theory). The structure of this filtered ${\mathbf F}_l$-space is completely known; see for example Section V of arXiv:0711.3878 (where your $l$ is called $p$).

If $K$ does not contain a primitive $l$-th root $\zeta$ of $1$, then put $K'=K(\zeta)$, $\Delta={\rm Gal}(K'|K)$ and $\omega:\Delta\to{\mathbf F}_l^\times$ the cyclotomic character giving the action of $G$ on the $l$-th roots of $1$. Then there is a natural bijection between the set of degree-$l$ cyclic extensions of $K$ and ${\mathbf F}_l$-lines in the $\omega$-eigenspace for the action of $\Delta$ on $K^{\prime\times}/K^{\prime\times l}$. The structure of this filtered ${\mathbf F}_l[\Delta]$-module can be completely determined; see for example arXiv:0912.2829.

If you are interested more generally in all degree-$l$ (separable) extensions of $K$ (and not just the cyclic ones), then something similar can be done. Put $L=K(\root{l-1}\of{K^\times})$ and $G={\rm Gal}(L|K)$. There is a natural bijection between the set of (isomorphism classes of) degree-$l$ (separable) extensions of $K$ and the set of $G$-stable lines in the ${\mathbf F}_l$-space $L^\times/L^{\times l}$. Again, the structure of this filtered ${\mathbf F}_l[G]$-module is completely known: see for example arXiv:1005.2016.

Finally, if you want to replace your $K$ by a finite extension of ${\mathbf F}_l((t))$, there are similar results using Artin-Schreier theory instead of Kummer theory. See for example arXiv:0909.2541 for degree-$l$ cyclic extensions and arXiv:1005.2016 for degree-$l$ separable extensions. They allow you to count the number of extensions with bounded ramification, of which there are only finitely many.

Here is a belated answer (I came across this question only today) which doesn't require anything more than Kummer theory --- or Artin-Schreier theory, if you want to allow $K$ to be a finite extension of ${\mathbf F}_l((t))$. I will confine myself to the more interesting case of degree-$l$ extensions (where $l$ is the residual characteristic).

If $K$ contains a primitive $l$-th root of $1$, then there is a natural bijection between the set of degree-$l$ cyclic extensions of $K$ and the set of ${\mathbf F}_l$-lines in $K^\times/K^{\times l}$ (Kummer theory). The structure of this filtered ${\mathbf F}_l$-space is completely known; see for example Section V of arXiv:0711.3878 (where your $l$ is called $p$).

If $K$ does not contain a primitive $l$-th root $\zeta$ of $1$, then put $K'=K(\zeta)$, $\Delta={\mathrm Gal(K'|K)$ and $\omega:\Delta\to{\mathbf F}_l^\times$ the cyclotomic character giving the action of $G$ on the $l$-th roots of $1$. Then there is a natural bijection between the set of degree-$l$ cyclic extensions of $K$ and ${\mathbf F}_l$-lines in the $\omega$-eigenspace for the action of $\Delta$ on $K^{\prime\times}/K^{\prime\times l}$. The structure of this filtered ${\mathbf F}_l[\Delta]$-module can be completely determined; see for example arXiv:0912.2829.

If you are interested more generally in all degree-$l$ (separable) extensions of $K$ (and not just the cyclic ones), then something similar can be done. Put $L=K(\root{l-1}\of{K^\times})$ and $G={\mathrm Gal}(L|K)$. There is a natural bijection between the set of (isomorphism classes of) degree-$l$ (separable) extensions of $K$ and the set of $G$-stable lines in the ${\mathbf F}_l$-space $L^\times/L^{\times l}$. Again, the structure of this filtered ${\mathbf F}_l[G]$-module is completely known: see for example arXiv:1005.2016.

Finally, if you want to replace your $K$ by a finite extension of ${\mathbf F}_l((t))$, there are similar results using Artin-Schreier theory instead of Kummer theory. See for example arXiv:0909.2541 for degree-$l$ cyclic extensions and arXiv:1005.2016 for degree-$l$ separable extensions.

Here is a belated answer (I came across this question only today) which doesn't require anything more than Kummer theory --- or Artin-Schreier theory, if you want to allow $K$ to be a finite extension of ${\mathbf F}_l((t))$. I will confine myself to the more interesting case of degree-$l$ extensions (where $l$ is the residual characteristic).

If $K$ contains a primitive $l$-th root of $1$, then there is a natural bijection between the set of degree-$l$ cyclic extensions of $K$ and the set of ${\mathbf F}_l$-lines in $K^\times/K^{\times l}$ (Kummer theory). The structure of this filtered ${\mathbf F}_l$-space is completely known; see for example Section V of arXiv:0711.3878 (where your $l$ is called $p$).

If $K$ does not contain a primitive $l$-th root $\zeta$ of $1$, then put $K'=K(\zeta)$, $\Delta={\rm Gal}(K'|K)$ and $\omega:\Delta\to{\mathbf F}_l^\times$ the cyclotomic character giving the action of $G$ on the $l$-th roots of $1$. Then there is a natural bijection between the set of degree-$l$ cyclic extensions of $K$ and ${\mathbf F}_l$-lines in the $\omega$-eigenspace for the action of $\Delta$ on $K^{\prime\times}/K^{\prime\times l}$. The structure of this filtered ${\mathbf F}_l[\Delta]$-module can be completely determined; see for example arXiv:0912.2829.

If you are interested more generally in all degree-$l$ (separable) extensions of $K$ (and not just the cyclic ones), then something similar can be done. Put $L=K(\root{l-1}\of{K^\times})$ and $G={\rm Gal}(L|K)$. There is a natural bijection between the set of (isomorphism classes of) degree-$l$ (separable) extensions of $K$ and the set of $G$-stable lines in the ${\mathbf F}_l$-space $L^\times/L^{\times l}$. Again, the structure of this filtered ${\mathbf F}_l[G]$-module is completely known: see for example arXiv:1005.2016.

Finally, if you want to replace your $K$ by a finite extension of ${\mathbf F}_l((t))$, there are similar results using Artin-Schreier theory instead of Kummer theory. See for example arXiv:0909.2541 for degree-$l$ cyclic extensions and arXiv:1005.2016 for degree-$l$ separable extensions.