# MA231: Abstract Algebra I

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 QR 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 - ai is called a splitting field of P(X).  If the field is the rational numbers Q and we consider a polynomial P(X) = X3 – 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.*

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.

Instructions: Please read the entire webpage for a concise rendering of information on fields, including various interesting examples.

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)

and then read “16.3 Ring Homomorphisms and Ideals,” pages 246-250.

use displayed on pages 410-417 on the PDF file.  The material linked
attributed to Thomas W. Judson and the original version can be
``````
• 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.

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)

use displayed on pages 410-417 on the PDF file.  The material linked
attributed to Thomas W. Judson and the original version can be
``````
• 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.

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.

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.