Unit 4: Fundamentals of Fields
We conclude this course with a unit on fields. Fields are general sets
with some defined operations of addition and multiplication that have
all the familiar properties of rational, real, and complex numbers.
They form commutative groups over both addition and multiplication and
have the distributive property that connects the two operations.
We will begin the unit with a formal definition of fields and consider
field properties. We will then look at some examples of fields and
learn about field extensions. An extension of some field *K** is
nothing more than a field M that contains K as a subfield.
Consider the real numbers R as an extension of the rational numbers
Q. R contains irrational numbers and, more importantly,
transcendental numbers. Transcendental numbers can never be solutions
to algebraic equations. Further, if there are algebraic and
transcendental extensions of fields in specific, what other kinds of
properties might be missing from one field F but included in some
minimal field containing F, in general?*
For instance, there are splitting fields. If P(X) is a polynomial over
some field *K, then any field **L containing K which P(X) can
“split” into linear factors X - a_{i} is called a splitting
field of P(X). If the field is the rational numbers Q and we
consider a polynomial P(X) = X^{3} – 7 with coefficients in the
rationals, one solution is irrational and two are complex, meaning that
we would need an extension with those solutions, at minimum. Any
extension that contains all three roots would be a splitting field.*
There is also the issue of P-closures. Suppose a field *K** does not
have a certain property P. Then any extension L containing K
that had property P would be called a P-closure for K. One example
is complex numbers being the algebraic closure for real numbers,
since some polynomials with real coefficients have complex roots. In
addition, an extension field of K is a separable closure for
K if it contains the smallest set of all possible finite separable
extensions of K within the algebraic closure.*
Unit 4 Time Advisory
This unit will take you 24 hours to complete.
☐ Subunit 4.1: 5 hours
☐ Subunit 4.2: 5 hours
☐ Subunit 4.3: 5 hours
☐ Subunit 4.4: 5 hours
☐ Subunit 4.5: 4 hours
Unit4 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Define and generate fields.
- Find an extension field given a field and polynomial solutions.
- Determine algebraic closure.
4.1 Definition and Properties of a Field
- Reading: Wolfram MathWorld: Eric. W. Weisstein’s “Field” and
“Field Axioms”
Link: Wolfram MathWorld: Eric W. Weisstein’s
“Field” (HTML) and
“Field Axioms”
(HTML)
Instructions: Please click on each link to read the information on
each webpage. The first webpage titled “Field” contains a concise
rendering of information on fields. It is very short, but provides
a good explanation in a few sentences of what a field is and is
not. The second webpage titled “Field Axioms” contains a concise
rendering of information on field axioms (properties). It is very
short, but provides a good explanation in a few sentences of what
makes a field a field.
Terms of Use: Please respect Wolfram MathWorld's terms of
use. MathWorld
webpages are free for academic use and may be hyperlinked, according
to their FAQ
site.
- Reading: Wikipedia: “Field (Mathematics)”
Link: Wikipedia: “Field
(Mathematics)”
(PDF)
Instructions: Please read the entire webpage for a concise rendering of information on fields, including various interesting examples.
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).
4.2 Extension Fields
- Reading: Stephen F. Austin State University: Thomas W. Judson’s
Abstract Algebra Theory and Applications: “Fields” and “Rings”
Link: Stephen F. Austin State University: Thomas W. Judson’s
Abstract Algebra Theory and Applications: “Fields” and
“Rings”
(PDF)
Also available in:
[EPUB](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2011/08/MA231-1.1.1book-Thomas-W.-Judson.epub)
Instructions: Please read “21.1 Extension Fields,” pages 329-340,
and then read “16.3 Ring Homomorphisms and Ideals,” pages 246-250.
Terms of Use: Please respect the copyright, license, and terms of
use displayed on pages 410-417 on the PDF file. The material linked
above is licensed under the [GNU Free Documentation
License](http://www.gnu.org/licenses/fdl.html) (HTML). It is
attributed to Thomas W. Judson and the original version can be
found [here](http://abstract.ups.edu/download/aata-20100827.pdf) (PDF).
Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Fields”: “Exercise Problem 2” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Fields”: “Exercise Problem 2” (PDF)
Also available in:Instructions: Work through problem 2 on page 350. Then, check the solution on page 407.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410-417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
4.3 Splitting Fields
- Reading: Stephen F. Austin State University: Thomas W. Judson’s
Abstract Algebra Theory and Applications: “Fields”
Link: Stephen F. Austin State University: Thomas W. Judson’s
Abstract Algebra Theory and Applications:
“Fields”
(PDF)
Also available in:
[EPUB](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2011/08/MA231-1.1.1book-Thomas-W.-Judson.epub)
Instructions: Please read “21.2 Splitting Fields,” pages 340-343.
Terms of Use: Please respect the copyright, license, and terms of
use displayed on pages 410-417 on the PDF file. The material linked
above is licensed under the [GNU Free Documentation
License](http://www.gnu.org/licenses/fdl.html) (HTML). It is
attributed to Thomas W. Judson and the original version can be
found [here](http://abstract.ups.edu/download/aata-20100827.pdf) (PDF).
Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Fields”: “Exercise Problem 3” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Fields”: “Exercise Problem 3” (PDF)
Also available in:
Instructions: Complete problem 3 on page 350, and then check your answer on page 407.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410-417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
4.4 Algebraic Closures
- Reading: Wolfram MathWorld: Eric W. Weisstein’s “Algebraic
Closure”
Link: Wolfram MathWorld: Eric W. Weisstein’s “Algebraic
Closure”
(HTML)
Instructions: Please click on the link to read the information on
the webpage for a summary of algebraic closure. It is very short
but provides a good explanation in a few sentences.
Terms of Use: Please respect Wolfram MathWorld's terms of
use. MathWorld
webpages are free for academic use and may be hyperlinked, according
to their FAQ
site.
- Reading: Wikipedia: “Algebraic Closure”
Link: Wikipedia: “Algebraic
Closure”
(PDF)
Instructions: Please read the entire webpage. This webpage contains a concise rendering of information on algebraic closure, including various examples. This topic is useful toward Abstract Algebra II.
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).
4.5 Separability
- Reading: Wolfram MathWorld: Eric W. Weisstein’s “Separable
Extension”
Link: Wolfram MathWorld: Eric W. Weisstein’s “Separable
Extension”
(HTML)
Instructions: Please click on the link to read the information on
the webpage for information on separable field extensions. It is
very short but provides a good explanation in a few sentences.
Terms of Use: Please respect Wolfram MathWorld's terms of
use. MathWorld
webpages are free for academic use and may be hyperlinked, according
to their FAQ
site.
- Reading: Wikipedia: “Separable Extensions”
Link: Wikipedia: “Separable
Extension”
(PDF)
Instructions: Please read the entire webpage for a concise rendering of information on separable field extensions, including various examples. This topic is useful toward Abstract Algebra II.
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).