Basic Notation for Field Extensions .... ....

[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 the meaning of some of the basic notation ...

In talking about ways to construct field extensions Lovett writes the following on page 322, Chapter 7 ... ... View attachment 6530In the above text from Lovett, he writes:

"... ... If \(\displaystyle \alpha \in R - F\), then \(\displaystyle F[ \alpha ]\) is the subring of \(\displaystyle R \) generated by \(\displaystyle R\) and \(\displaystyle \alpha\).** See Section 5.2.1. Since \(\displaystyle F[ \alpha ] \) is an integral domain, we can take the field of fractions \(\displaystyle F( \alpha )\) of \(\displaystyle F[ \alpha \)]. See Section 6.2. ... ... "

(** see end of post ...)My problem is that the way Lovett seems to define \(\displaystyle F[ \alpha ]\) and \(\displaystyle F[ \alpha ]\) seems (on the surface, at least) to be different from all the other texts I am reading ... for example Dummit and Foote or Gallian ...

Lovett's definition of the notation \(\displaystyle F[ \alpha ]\) comes in sections 5.2.1 and 5.2.2 where he defines rings generated by elements as follows ... ... View attachment 6531He goes on from the above in the next section to define rings of polynomials using the same notation as he did for generated subrings ... as follows:View attachment 6532
Lovett defines \(\displaystyle R(x)\) in Section 6.2.2 as the field of fractions of \(\displaystyle R[x]\) ... as follows:View attachment 6533Dummit and Foote, on the other hand (like a number of other algebra texts) define \(\displaystyle F( \alpha )\) as follows: View attachment 6534

Further, Dummit and Foote simply define \(\displaystyle R[x]\) in terms of polynomial rings ... as follows:View attachment 6535

Can someone please give an explanation of the apparently different definitions between Lovett and the other texts on this subject ...
[Please excuse me moving between rings and fields in the above definitions ... ] Hope someone can help ...

Peter
NOTE** Where Lovett writes

" ... ... If \(\displaystyle \alpha \in R - F\), then \(\displaystyle F[ \alpha ]\) is the subring of \(\displaystyle R\) generated by \(\displaystyle R\) and \(\displaystyle \alpha\)."

Is this correct?

Should this read:" ... ... If \(\displaystyle \alpha \in R - F\), then \(\displaystyle F[ \alpha ]\) is the subring of \(\displaystyle R\) generated by \(\displaystyle F\) and \(\displaystyle \alpha\)."

 

[Euge]

Perhaps the following will clarify things a bit. Let $K/F$ be a field extension. Given a subset $S$ of $K$, define $F$ to be smallest subring of $K$ containing $F$ and $S$, and $F(S)$ the smallest subfield of $K$ containing $F$ and $S$. Given $\alpha_1,\ldots \alpha_m\in K$, it can be shown that

1. $F[\alpha_1,\ldots, \alpha_m]$ consists of all evaluations $f(\alpha_1,\ldots, \alpha_m)$, where $f$ is a polynomial over $F$ in $m$ variables.

2. The elements of $F(\alpha_1,\ldots, \alpha_m)$ are quotients of the form $$\frac{p(\alpha_1,\ldots,\alpha_m)}{q(\alpha_1,\ldots, \alpha_m)}$$ where $p$ and $q$ are polynomials over $F$ in $m$ variables with $q(\alpha_1,\ldots, \alpha_m) \neq 0$. In other words, $F(\alpha_1,\ldots, \alpha_m)$ is the field of fractions of $F[\alpha_1,\ldots, \alpha_m]$.

Your correction at the end is right one, as $F[\alpha]$ is indeed the subring of $R$ generated by $F$ and $\alpha$.

 

[Math Amateur]

Perhaps the following will clarify things a bit. Let $K/F$ be a field extension. Given a subset $S$ of $K$, define $F$ to be smallest subring of $K$ containing $F$ and $S$, and $F(S)$ the smallest subfield of $K$ containing $F$ and $S$. Given $\alpha_1,\ldots \alpha_m\in K$, it can be shown that

1. $F[\alpha_1,\ldots, \alpha_m]$ consists of all evaluations $f(\alpha_1,\ldots, \alpha_m)$, where $f$ is a polynomial over $F$ in $m$ variables.

2. The elements of $F(\alpha_1,\ldots, \alpha_m)$ are quotients of the form $$\frac{p(\alpha_1,\ldots,\alpha_m)}{q(\alpha_1,\ldots, \alpha_m)}$$ where $p$ and $q$ are polynomials over $F$ in $m$ variables with $q(\alpha_1,\ldots, \alpha_m) \neq 0$. In other words, $F(\alpha_1,\ldots, \alpha_m)$ is the field of fractions of $F[\alpha_1,\ldots, \alpha_m]$.

Your correction at the end is right one, as $F[\alpha]$ is indeed the subring of $R$ generated by $F$ and $\alpha$.

Thanks for the help, Euge ...

Peter