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?
If you understood the above this last must be easier now to attack.
Originally Posted by Horizon