Let π be a ring.
A polynomialπ(π₯) in the indeterminateπ₯ and with coΓ«fficients in π is a finite linear combination of nonnegative powers of π₯ with coΓ«fficients in π :12ring
π(π₯)=ββπ=0πππ₯π
where π(β) has finite support3,
hence it may be viewed as an element of the free moduleπ [β0].
This free module forms the polynomial ringπ [π₯] with the structure of a ring (and K-monoid) given by the Monoid ring construction, thus
π₯πβ π₯π=π₯π+π
The leading term of a polynomial is the term πππ₯π with the largest exponent π,
and the coΓ«fficient ππ is called the degreedegβ‘π.
We write degβ‘0=ββ.
A polynomial with leading coΓ«fficient one is called a monic polynomial (not to be confused with monic).
A polynomial is irreducible if has no divisors other than itself and 1 (similar to prime numbers),
however a polynomial can often be reduced by looking at a bigger underlying ring,
for examples π₯2+1 can only be factorised using the complex numbers.
A polynomial in multiple indeterminates may be formed by iterating the above process, so π [π₯,π¦,π§]=π [π₯][π¦][π§].
Universal property
An fundamental property of a polynomial ring is that the elements {π₯π}βπ=1 are in the centre.
The polynomial ring π [π₯] is characterized up to unique isomorphism by the following universal property:
Let π be a ring. The polynomial ring is a pair consisting of a ring π [π₯] and a ring homomorphismππ :π βπ [π₯] such that π₯ is an element of the centreπ(π [π₯]) and given any ring homomorphism π:π βπ and element π of the centralizerπΆ(π(π )), then there exists a unique ring homomorphism Β―π:π [π₯]βπ such that the following diagram commutes
and Β―π(π₯)=π.
Evaluation map
Let π(π₯)βπ [π₯] and πβπ .
By the above construction,
there exists a unique ring homomorphism π(π):π [π₯]βπ such that π(π)ππ =idπ and π(π)(π₯)=π,
which is called the evaluation map at π.