Irreducible polynomial, Primitive Polynomial and Minimal Polynomial

Question

I read about irreducible polynomial, primitive polynomial and minimal polynomial and now i am not able to differentiate between them, its chaos in my mind. Can somebody describe what they are actually and what is the difference between them.

Secondly how do we generate irreducible polynomials. Explanation in simple terms will be more useful, actually i am computer science student.

Answer 1

• Start with $k = \mathbb{F}_p$ with $p$ prime. Then $k[x]$ is the ring of polynomials with coefficients in $k$, and $P(x) = \sum_{m=0}^n c_m x^m$ is irreducible means there is no polynomials $Q,R \in k[x]$ with $1 \le deg(Q) < deg(P)$ such that $P = Q R$. Now $P$ has some roots in some larger field, and $P \in k[x]$ is the minimal polynomial of $\alpha$ iff $P(\alpha) = 0$ and $P$ is irreducible in $k[x]$.

• You need to look at the field extensions $k(\alpha)$ with $\alpha$ algeabraic over $k$. The main theorem is that $k(\alpha)$ is isomorphic to $k[x]/(P)$ where $P$ is the minimal polynomial of $\alpha$.

  Conversely, $L = k[x]/(P)$ is a field whenever $P$ is irreducible (otherwise $L$ has some zero-divisors).

• Also there is at most one finite field with $q$ elements, so $k(\alpha)$ is denoted as $\mathbb{F}_q$ (where $q = p^n, n = deg(P)$)

• $k(\alpha) = \mathbb{F}_q$ is a finite field, thus its multiplicative group is cyclic. Let $\beta$ be a generator of $\mathbb{F}_q^\times$ and $Q$ its minimal polynomial. Then $\beta$ is a $p^n-1$th primitive root of unity in $\overline{\mathbb{F}_p}$ (the algebraic closure) and $Q$ is called a primitive polynomial.

• You need to take some examples, and look at the number fields where $\mathbb{F}_p$ is replaced by $\mathbb{Q}$

• For generating primitive polynomials, I don't know, but for generating the minimal polynomial of an element $\alpha$ you can use the Galois conjugates

Answer 2

Irreducible Polynomial: A polynomial that can't be factored anymore, otherwise the coefficients won't be in GF(p). Ex. x^2+x+1

Minimal Polynomial (of some element $\alpha$ not in GF(p): The lowest-degree irreducible polynomial, where $\alpha$ is a root, and where the coefficients are all in GF(p). Ex. x^2+1 is the minimal polynomial of i.

Primitive Polynomial: The minimal polynomial of a primitive element (a generator of the multiplicative group) of a finite field. Ex. If $\alpha$ is the root of x^2+x+1, and x^2+x+1 is primative, then $\alpha^n$ will generate all elements of the multiplicative group of a finite field. This is true actually, as {1,$\alpha$, $\alpha^2$, $\alpha^3$} (plugging $\alpha$ into x^2+x+1) yields {1,$\alpha$,$\alpha+1$,1}, which are all the elements of the multiplicative group of GF(2^2).

Change variables from $\alpha$ to $x$, this becomes also equivalent to saying that P(x) is a primitive polynomial, if the finite field GF(p^n) constructed by GF(p)[n]/P(x), has x as a generator.