Irreducible polynomials over a field F play a similar, in a certain sense, role in F[x] to the role
the prime numbers play for Z. Using your favorite proof of the existence of an infinite number of
primes as inspiration, please prove that there are infinitely many irreducible polynomials over a
field F.
Solution
if there were finite such polynomials.... p1.p2.p3.........pn
then
p1 * p2 * p3 * ......pn +1 (multiplicative unity) is a polynomial
which is either irreducible or can be written as a product of irreducible polynomials.
It cannot be irreducible since it is different from the set of irreducibles.
Also it cannot be divisible by any irreducible . Say it does pi divides this new polynomial fr
some i , then
pi divides p1*p2*......pn + 1
and also pi is one of p1,p2....pn so divides p1*p2*....pn
hence it divides 1
contradiction.
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Irreducible polynomials over a field F play a similar, in a certain .pdf
1. Irreducible polynomials over a field F play a similar, in a certain sense, role in F[x] to the role
the prime numbers play for Z. Using your favorite proof of the existence of an infinite number of
primes as inspiration, please prove that there are infinitely many irreducible polynomials over a
field F.
Solution
if there were finite such polynomials.... p1.p2.p3.........pn
then
p1 * p2 * p3 * ......pn +1 (multiplicative unity) is a polynomial
which is either irreducible or can be written as a product of irreducible polynomials.
It cannot be irreducible since it is different from the set of irreducibles.
Also it cannot be divisible by any irreducible . Say it does pi divides this new polynomial fr
some i , then
pi divides p1*p2*......pn + 1
and also pi is one of p1,p2....pn so divides p1*p2*....pn
hence it divides 1
contradiction
![Irreducible polynomials over a field F play a similar, in a certain sense, role in F[x] to the role
the prime numbers play for Z. Using your favorite proof of the existence of an infinite number of
primes as inspiration, please prove that there are infinitely many irreducible polynomials over a
field F.
Solution
if there were finite such polynomials.... p1.p2.p3.........pn
then
p1 * p2 * p3 * ......pn +1 (multiplicative unity) is a polynomial
which is either irreducible or can be written as a product of irreducible polynomials.
It cannot be irreducible since it is different from the set of irreducibles.
Also it cannot be divisible by any irreducible . Say it does pi divides this new polynomial fr
some i , then
pi divides p1*p2*......pn + 1
and also pi is one of p1,p2....pn so divides p1*p2*....pn
hence it divides 1
contradiction](https://image.slidesharecdn.com/irreduciblepolynomialsoverafieldfplayasimilarinacertain-230707221153-66c2a39d/85/Irreducible-polynomials-over-a-field-F-play-a-similar-in-a-certain-pdf-1-320.jpg)
