For a field extension of the field , with a transcendental element over , my textbook says the following:
" contains the field of fractions of , which is the smallest subfield containing and , we denote this field by ."
It also says that is defined to be the image of the evaluation homomorphism . I think this corresponds to the usual notation of meaning polynomials in , with coefficients form , right?
Anyway, my first question is how do we know that is the smallest field containing and ?
Next, in an example it says that is transendental in , the field is isomorphic to the field of rational functions over in the indeterminate .
This is very confusing. How does it conclude this? by the definition should equal the fraction field of . If is a field then we can conclude that its fraction field is itself, and by the evaluation homomorphism above (with codomain restricted to its image, making it an isomorphism), we do see , which is almost the result, I think does actually equal , but I'm not sure why, can anyone explain? Also, I don't see how is a field, in fact I think it's only a integral domain.
So can someone explain how this example works?
Also, in the exercise section it asks, for a indeterminate whether the statement: is true or false. It should be similar to the above example, can someone explain this too?