Ultraproducts and Los' Theorem

Fill the blank to complete the following definition of an ultrafilter.

An ultrafilter on a set I is a filter on I that is (a)   with respect to set inclusion.

A principal ultrafilter on I is an ultrafilter U such that.

Los's theorem states that, given a formula $$\varphi\left(x_1,\dots,x_n\right)$$ , any tuple $$\overline{a}=\left(a^1,\dots,a^n\right)$$  in an ultraproduct $$M^{\ast}=\prod_{i\in I}^{ }M_i\ /U$$ and representatives $$\left(a_i^j\right)_i\in\prod_{i\in I}^{ }M_i$$   for the $$a^j$$ , then

$$M^{\ast}\models\varphi\left(\overline{a}\right)$$   some iff...

Which of the properties of filters is used in the existential case of the proof of Los' theorem?

What kind of proof is used in Los' Theorem.

(a)  

Is the following statement true?:

Given a structure $$M$$  and a principal ultrapower $$M^{\ast}$$   of M, M is isomorphic to $$M^{\ast}$$ .

Is the following statement true?:

Given a structure $$M$$  and an ultrapower $$M^{\ast}$$   of M, M is isomorphic to $$M^{\ast}$$ .

Is the following statement true?:

Given a structure $$M$$  and an ultrapower $$M^{\ast}$$   of M, M embeds elementarily into $$M^{\ast}$$ .

Is the following statement true?:

Given some structures $$\left(M_i\right)_{i\in I}$$  and an ultraproduct $$M^{\ast}=\prod_{i\in I}^{ }M_{i\ }/U$$  , each component $$M_i$$   embeds elementarily into $$M^{\ast}$$ .

Is the following statement true?:

Given some field structures $$\left(M_i\right)_{i\in I}$$  in the language of rings and an ultraproduct $$M^{\ast}=\prod_{i\in I}^{ }M_{i\ }/U$$  , $$M^{\ast}$$ must also be a field.