Question

What is a prime polynomial ?

Answer 1

The term not factorable tends to be used, but essentially they are primes...

Explanation:

Warning: Long answer.

This question is more interesting than you might think.

The quick answer is that we tend to call them not-factorable rather than prime, but the irreducible factors are primes in the ring of polynomials.

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What is a ring?

A ring is a set `R` with addition (`+`) and multiplication (`*`) operations with (at least some of) the expected properties:

• `R` is closed under addition:
  `color(white)(1/1)` If `a, b in R` then `a+b in R`

• Addition is commutative:
  `color(white)(1/1)` `a+b = b+a` for all `a, b in R`

• Addition is associative:
  `color(white)(1/1)` `a+(b+c) = (a+b)+c` for all `a, b, c in R`

• There is an identity `0` under addition:
  `color(white)(1/1)` `a+0 = 0+a = a` for all `a in R`

• Every element has an inverse under addition:
  `color(white)(1/1)` If `a in R` then there is `b in R` such that `a+b=0`

• `R` is closed under multiplication:
  `color(white)(1/1)` If `a, b in R` then `a*b in R`

• Multiplication is associative:
  `color(white)(1/1)` `a*(b*c) = (a*b)*c` for all `a, b, c in R`

• There is an identity `1` under multiplication:
  `color(white)(1/1)` `a*1 = 1*a = a` for all `a in R`

• Multiplication is left and right distributive over addition:
  `color(white)(1/1)` `a*(b+c) = (a*b)+(a*c)` for all `a, b, c in R`
  `color(white)(1/1)` `(a+b)*c = (a*c)+(b*c)` for all `a, b, c in R`

If multiplication is also commutative then `R` is called a commutative ring.

The most important example of a commutative ring is the integers `ZZ`.

`color(white)()`
Polynomials over a ring

If `R` is any ring, then we can introduce a variable `x` and consider the set `R[x]` of polynomials in `x` with coefficients being elements of `R`. For example, `ZZ[x]` being the set of polynomials with integer coefficients.

These polynomials form a ring under the normal addition and multiplication operations for polynomials.

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Prime element of a ring

An element `a in R` is said to be divisible by `b in R` if and only if there is another element `c in R` such that:

`a = b*c`

An element `u in R` is called a unit of `R` if and only if it has a multiplicative inverse in `R`. For example in `ZZ` the only units are `1` and `-1`.

An element `a in R` is called prime if and only if for all `b, c in R` we find:

`b*c = a" " => " at least one of "b" or "c" is a unit"`

That is `a` is only divisible by units and unit multiples of itself.

Then a polynomial `P in R[x]` (i.e. a polynomial in `x` with coefficients in `R`) is factorable over `R` if and only if it is not prime.