When does a field of for which a set of matrices is defined on become important?

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I have already gone through several linear algebra related courses without any notion of a field and I am having trouble reading a serious linear algebra book that talks about infinite field, so and so forth.

Obviously then, the notion of a field has little practical value in applied linear algebra. But my question is when or under what context do we MUST mention the field of which something is defined on? Can someone show me an elementary example for my small brain?