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Today I found an exercise that asked to demonstrate that every field $F/K$ extension generated by elements of degree 2 is normal.

If the extension were finitely generated, let's say $F=K(\alpha_1,\dots,\alpha_n)$, it is simple to prove as we can see that $F$ is the splitting field of $g_1(X)\dots g_n(X)$ where $g_i(X)=Irr(\alpha_i,K)$ (monical irreducible polynomial with $\alpha_i$ as a root).

But, is the statement of the exercise true for infinitely generated extensions?

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Yes. The general statement is that $F/K$ is a normal extension if and only if $F$ is the splitting field of a collection of polynomials over $K$. In the finite case this is equivalent to $F$ being a splitting field of a single polynomial. (and so many Galois theory books only mention this case)

In your case, take the collection to be the minimal polynomials of all generators. They split because they are all of degree $2$.

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