Daily Prep 20

Section Daily Prep 20

Today we will discuss particularly nice irreducible polynomials called primitive polynomials, whose roots are primitive elements. These are ideal for constructing finite fields, and their properties will be very useful in studying our next class of codes.

🔗

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 exponent of a polynomial, primitive polynomial.
🔗

🔗
State and apply the result that the exponent of an irreducible polynomial determines the multiplicative order of its roots.
🔗

🔗

🔗

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.

🔗

Count the number of monic primitive polynomials of degree \(n\) over \(\GF(q)\text{.}\)
🔗

🔗
State the definition and give examples of cyclotomic cosets.
🔗

🔗
Compute cyclotomic cosets and factor polynomials of the form \(x^e-1\) for \(\gcd(e,q)=1\) over \(\GF(q)\)
🔗

🔗

🔗

Subsection Resources for Learning

Use these resources to prepare for class and answer the questions below.

Roth, Sections 7.4-7.5, pp.227-232
🔗

🔗

🔗

Figure 61. Reference Video for Exponents & Primitive Polynomials

🔗

Subsection Important Terms

Definition 62. Exponent of a Polynomial.

The exponent or order of a polynomial \(a(x)\in F[x]\) with \(\gcd(a(x),x)=1\) is the smallest positive integer \(e\) such that \(a(x)\mid x^e-1\text{,}\) denoted \(\exp(a(x))\text{.}\)

🔗

Definition 63. Primitive Polynomial.

An irreducible polynomial of degree \(n\) in \(\GF(q)[x]\) is called a primitive polynomial if its exponent is \(q^n-1\text{.}\)

🔗

Proposition 64. The Exponent of an Irreducible is the Order of its Roots.

Let \(a(x)\in F[x]\) be an irreducible polynomial which is not a constant multiple of \(x\text{.}\) Then \(\exp(a(x))\) is the order of any root of \(a(x)\) in an extension field of \(F\text{.}\)

🔗

Proposition 65. Relation between Order of Root and Minimum Polynomial Degree.

Let \(\alpha\neq 0\) have order \(e\) in an extension field of \(F\text{.}\) Then the degree of the minimum polynomial of \(\alpha\) over \(F\) is the smallest positive integer \(m\) such that \(e\mid q^m-1\text{.}\)

🔗

Prev Top Next