The AKE priciple states that two finitely ramified Henselian field (this means that the field is either of residual caracteristic 0 or is of characteristic $p$ and there is only a finite number of elements x of the value group such that $0< x < v(p)$) are elementary equivalent if and only if their value groups are elementary equivalent (in the language of group) and their residue fields elementary equivalent (in the language of ring).

I'm looking for application of this principle to prove non trivial results.

Thanks to any one who might answer !

The AKE priciple states that two finitely ramified Henselian field (this means that the field is either of residual caracteristic 0 or is of characteristic $p$ and there is only a finite number of elements x of the value group such that $0< x < v(p)$) are elementary equivalent if and only if their value groups are elementary equivalent (in the language of group) and their residue fields elementary equivalent (in the language of ring).

I'm looking for application of this principle to prove non trivial results.

I also have a side question which I fear is a dumb one. Being algebraically closed is a first order property. So it should be equivalent to properties on the residues field and on the value group. What could be these properties ?

Thanks to any one who might answer !

The AKE priciple states that two finitely ramified Henselian field (this means that the field is either of residual caracteristic 0 or is of characteristic $p$ and there is only a finite number of elements x of the value group such that $0< x < v(p)$) are elementary equivalent if and only if their value groups are elementary equivalent (in the language of group) and their residue fields elementary equivalent (in the language of ring).

I'm looking for application of this principle to prove non trivial results.

I also have a side question which I fear is a dumb one. The property of being algebraically closed is first order so according to the AKE principle it is equivalent to properties on the value group and properties on the residue field. Which proprty is it ?

Thanks to any one who might answer !

The AKE priciple states that two finitely ramified Henselian field (this means that the field is either of residual caracteristic 0 or is of characteristic $p$ and there is only a finite number of elements x of the value group such that $0< x < v(p)$) are elementary equivalent if and only if their value groups are elementary equivalent (in the language of group) and their residue fields elementary equivalent (in the language of ring).

I'm looking for application of this principle to prove non trivial results.

Thanks to any one who might answer !