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Section Daily Prep 11
Today we will continue our investigation of construction and computation and properties of non-prime order finite fields.
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 the definition and give examples of: primitive element, extension field, extension degree.
Carry out addition and multiplication in fields defined by reducing polynomials over a field modulo an irreducible polynomial.
Compute the correspondence between powers of a primitive element in
\(F[x]/p(x)\) and the polynomial representations of those elements.
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.
State the definition and give examples of: root, multiplicity, simple root, characteristic, prime subfield.
State and apply the Degree Mantra,
[provisional cross-reference: qth Power Theorem],
[provisional cross-reference: Multiplicative Groups of Finite Fields are Cyclic Theorem]
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Guruswami, Rudra, & Sudan, Sections 5.1, D.5, pp. 93-95, 528-532
Roth, Sections 3.3-3.5, pp. 56-62
Vanstone & van Oorschot, Sections 2.1-2.3, pp. 23-32
Subsection Important Terms
Definition 45 . Primitive Element.
An element \(\alpha\) of a finite field \(F\) is primitive if it generates the multiplicative group of the field, i.e.
\begin{equation*}
\{ \alpha^i \mid i \in \Z \} = F\setminus\{0\}\text{.}
\end{equation*}
Definition 46 . Extension Field/Subfield.
Let
\(F\) and
\(E\) be fields. We say
\(E\) is an
extension field of
\(F\) and
\(F\) is a
subfield of
\(E\) if
\(F\subseteq E\) and the operations of
\(E\) restricted to elements of
\(F\) agree with the operations of
\(F\text{.}\)
Definition 47 . Extension Degree.
Let
\(E\) be an extension field of
\(F\text{.}\) The
extension degree of
\(E\) over
\(F\text{,}\) denoted
\([E:F]\text{,}\) is the vector space dimension of
\(E\) as a vector space over
\(F\text{.}\)
