- #1
Math Amateur
Gold Member
MHB
- 3,998
- 48
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 ... ...
In the above text from Lovett, he writes:
"... ... If ##\alpha \in R - F##, then ##F[ \alpha ]## is the subring of ##R## generated by ##R## and ##\alpha##.** See Section 5.2.1. Since ##F[ \alpha ]## is an integral domain, we can take the field of fractions ##F( \alpha )## of ##F[ \alpha ]##. See Section 6.2. ... ...
(** See note at end of post ... )"My problem is that the way Lovett seems to define ##F[ \alpha ]## and ##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 ##F[ \alpha ]## comes in sections 5.2.1 and 5.2.2 where he defines rings generated by elements as follows ... ...
He 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:
Lovett defines ##R(x)## in Section 6.2.2 as the field of fractions of ##R[x]## ... as follows:
Dummit and Foote, on the other hand (like a number of other algebra texts) define ##F( \alpha )## as follows:
Further, Dummit and Foote simply define ##R[x]## in terms of polynomial rings ... as follows:
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 ##\alpha \in R - F##, then ##F[ \alpha ]## is the subring of ##R## generated by ##R## and ##\alpha##."
Is this correct?
Should this read:" ... ... If ##\alpha \in R - F##, then ##F[ \alpha ]## is the subring of ##R## generated by ##F## and ##\alpha##."
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 ... ...
In the above text from Lovett, he writes:
"... ... If ##\alpha \in R - F##, then ##F[ \alpha ]## is the subring of ##R## generated by ##R## and ##\alpha##.** See Section 5.2.1. Since ##F[ \alpha ]## is an integral domain, we can take the field of fractions ##F( \alpha )## of ##F[ \alpha ]##. See Section 6.2. ... ...
(** See note at end of post ... )"My problem is that the way Lovett seems to define ##F[ \alpha ]## and ##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 ##F[ \alpha ]## comes in sections 5.2.1 and 5.2.2 where he defines rings generated by elements as follows ... ...
He 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:
Dummit and Foote, on the other hand (like a number of other algebra texts) define ##F( \alpha )## as follows:
Further, Dummit and Foote simply define ##R[x]## in terms of polynomial rings ... as follows:
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 ##\alpha \in R - F##, then ##F[ \alpha ]## is the subring of ##R## generated by ##R## and ##\alpha##."
Is this correct?
Should this read:" ... ... If ##\alpha \in R - F##, then ##F[ \alpha ]## is the subring of ##R## generated by ##F## and ##\alpha##."
Attachments
-
Lovett - Field Extension - page 322 ... ....png52.2 KB · Views: 531
-
Lovett - Section 5.2.1 - Generated Subrings ... ....png25 KB · Views: 563
-
Lovett - Section 5.2.2 - Polynomial Rings ... ....png18.9 KB · Views: 506
-
Lovett - Example 6.2.11 - Rational Expressions ... ....png10.9 KB · Views: 516
-
Lovett - Example 6.2.11 - Rational Expressions ... ....png10.9 KB · Views: 458
-
D&F - Field Extensions - page 517.png17.7 KB · Views: 452
-
D&F - Defn of R[X] as ring of polynomials ....png30.8 KB · Views: 542
Last edited: