Table of Contents
How do you determine if a polynomial is irreducible over a finite field?
Irreducible polynomials Let F be a finite field. As for general fields, a non-constant polynomial f in F[x] is said to be irreducible over F if it is not the product of two polynomials of positive degree. A polynomial of positive degree that is not irreducible over F is called reducible over F.
What happens to irreducible polynomials?
Every polynomial over F may be decomposed into the product of a non-zero constant and a finite number of non-constant irreducible primitive polynomials. The non-zero constant may itself be decomposed into the product of a unit of F and a finite number of irreducible elements of F.
Are irreducible polynomials minimal?
A minimal polynomial is irreducible. Let E/F be a field extension over F as above, α ∈ E, and f ∈ F[x] a minimal polynomial for α. Suppose f = gh, where g, h ∈ F[x] are of lower degree than f. Thus minimal polynomials are irreducible.
How do you prove a field is finite?
A finite field of order q exists if and only if q is a prime power pk (where p is a prime number and k is a positive integer). In a field of order pk, adding p copies of any element always results in zero; that is, the characteristic of the field is p.
Does irreducible mean no roots?
If a polynomial with degree 2 or higher is irreducible in , then it has no roots in . If a polynomial with degree 2 or 3 has no roots in , then it is irreducible in .
Why is a polynomial irreducible?
A polynomial is said to be irreducible if it cannot be factored into nontrivial polynomials over the same field.
How do you find irreducible polynomials?
Use long division or other arguments to show that none of these is actually a factor. If a polynomial with degree 2 or higher is irreducible in , then it has no roots in . If a polynomial with degree 2 or 3 has no roots in , then it is irreducible in .
Why finite fields are used in cryptography?
A Finite Field denoted by Fp, where p is a prime number, works well with cryptographic algorithms like AES, RSA , etc. because of the following reasons: We need to decrypt the encrypted message, this is only possible when a unique (bijective) inverse of a function is available.
Why are finite fields prime?
Since F is finite, it must have characteristic p for some prime p (by Corollary 2.19). So, all finite fields must have prime power order – there is no finite field with 6 elements, for example. If f ∈ Fp[x] is irreducible of degree n over Fp, then adjoining a root of f to Fp yields a finite field of pn elements.