Add new constants $\bar c$ and consider
$$
\operatorname{ElDiag}(M)\cup\{\varphi(\bar c):\varphi\in p\}.
$$
Every finite subset is satisfiable because $p$ is consistent over $M$. By the <compactness theorem> it has a model $N'$. The constants naming $M$ give an elementary embedding $M\to N'$, and $\bar c^{N'}$ realizes $p$. Replacing $N'$ by an isomorphic copy containing $M$ gives the required <realization of a type in an elementary extension>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 120 / 2 / b / Solution

Every complete type over parameters from $M$ is realized in some elementary extension of $M$. Apply compactness to the elementary diagram of $M$ together with a new tuple of constants satisfying the type.


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