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Section Daily Prep 12
Today we will discuss splitting fields for polynomials over finite fields and see how to use them to establish the existence of finite fields of all prime power sizes. We will then use the field
\(GF(16)\) to extend the
\([15,11,3]_2\) Hamming code to a
\(2\) -error correcting code.
Subsection Learning Objectives
Subsubsection Basic Learning Objectives
Objectives
Before our class meeting, you should use the resources below to be able to learn the following. You should be reasonably fluent with these; we’ll answer some questions on them in class but not reteach them in detail.
State and give examples of the definition of: field in which a polynomial splits, splitting field, formal derivative.
Explain why splitting fields always exist.
Explain why polynomials have repeated roots if and only if they have a common factor with their formal derivative.
Subsubsection Advanced Learning Objectives
Objectives
During our class meeting, we will work on learning the following. Fluency with these is not expected or required before class.
Identify a finite field of size
\(p^n\) as the roots of a polynomial in its splitting field.
Implement the construction of double-error-correcting (narrow-sense) binary alternant codes.
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Roth, Sections 3.7-3.8, pp. 64-70
MacWilliams & Sloane, Sections 3.1-3.3, pp. 80-88
Subsection Important Terms
