Field Extensions and "Free Parameters" ?

[Math Amateur]

I am reading "Abstract Algebra: Structures and Applications" by Stephen Lovett ...

I am currently focused on Chapter 7: Field Extensions ... ...

I need help with some remarks of Lovett pertaining to Theorem 7.1.10 ...The statement of Theorem 7.1.10 reads as follows (page 325) :View attachment 6628The remarks pertaining to Theorem 7.1.10 read as follows (page 326) :View attachment 6629I do not understand the use of 't' ... nor do I fully understand the analysis involving it ...


My specific question is as follows:In the above remarks, Lovett writes the following:

" ... ... For example keeping \(\displaystyle t\) as a free parameter, \(\displaystyle F[t]\) is a subring of \(\displaystyle F(t)\). ... ... "What is '\(\displaystyle t\)' and why exactly are we introducing it?

Why not stick with \(\displaystyle x\) and \(\displaystyle F[x]\) and \(\displaystyle F(x)\) ... ... ?

I note that Lovett does not usually introduce a "free parameter" (whatever that is?) and happily deals with the indeterminate \(\displaystyle x\) ... ... ?? ... ... so ... indeed, one may ask when is a "free parameter" necessary and when is it not needed ... ?

Hope someone can help ...

Peter

 

[jvicens]

Dear Peter,

In abstract algebra, we often use letters such as x, y, or z as variables or indeterminates to represent elements in a ring or field. These letters are usually chosen arbitrarily and can be replaced with any other letter without changing the underlying mathematical concepts. In the context of field extensions, we use the letter t to represent an indeterminate in the polynomial ring F[t]. This is because t is commonly used to represent an element in the extension field F(t).

The reason for introducing t as a free parameter is to show that F[t] is a subring of F(t), which means that it contains all the elements of F(t) and is closed under addition and multiplication. This is an important result in field theory and helps us understand the structure of field extensions.

As for your question about when a free parameter is necessary, it depends on the specific problem at hand. In this case, introducing t allows us to make a clear distinction between the polynomial ring F[t] and the field of rational functions F(t), which can be helpful in understanding their relationship.

I hope this helps clarify the use of t in Theorem 7.1.10. Feel free to ask any further questions or for clarification.