Why do we study irreducibility of polynomials? What are some real-life applications?

Question

Irreducibility of polynomial functions seems like one of the major topics in introductory abstract algebra, and there are many theorems (such as Eisenstein's Criterion, etc) on testing whether a polynomial is reducible in $\mathbb{Q}[x], \mathbb{R}[x],$ and $\mathbb{C}[x]$, etc.

I wonder, why do we care so much about irreducibility? What are some of the applications (preferably in engineering or physics, but other areas are also great)?

Thanks!

Answer 1

Polynomial codes are error detection and correction codes that start by choosing a finite field of prime power order, $GF(p^n)$. The easiest way to explicitly construct such a field in which you can actually compute is to construct a non-prime field. For instance, if $P(X)$ is an irreducible polynomial of degree $n> 1$, then $GF(p^n) = GF(p)[X]/P(X)$ is a finite field of order $p^n$. (It doesn't matter which irreducible polynomial of degree $n$ you use -- you get the same field, but the names of some of the elements seem different.)

Polynomial codes include

• cyclic codes, which give cyclic redundancy checks(CRCs). iSCSI and SCTP have incorporated CRCs.;
• BCH codes and Reed-Solomon codes, which are used in radio communication, CDs, DVDs, HDDs, SSDs, some 2-d barcodes, e.g. QR codes, data transmission and reconstruction, and other forms of data storage and retrieval.