Splitting Fields - Dummit and Foote - Exercise 1, page 545

Dummit and Foote Exercise 1 on page 545 reads as follows:

Determine the splitting field and its degree over for .

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I have started on the solution to this exercise as follows:

The two roots of are and

Thus the splitting field is

We note that since the product of and i must be in

The element is algebraic over and so is the minimal polynomial for over .

Thus the degree of over is the degree of = 4

Hence = 4

The element is also algebraic over and so is the minimal polynomial for over .

Thus the degree of over is the degree of = 4

Hence = 4

But where to from here - need to find the degree of the splitting field.

Can someone please confirm that my reasoning above is valid and show me the way forward from here?

Peter