Polynomial Ring

Extension from Elementary Algebra §

• We say to mean the set of all polynomials with coefficients in the ring . is a ring called the Polynomial Ring where addition and multiplication are defined similarly for polynomials.

  More formally, a Polynomial with coefficients in is an infinite formal sum

  Where and for all but a finite number of values of . We refer to as the coefficients. We refer to as the indeterminate.

  If for some , , the largest such is the degree of denoted .

  If all , then .

  All polynomials are written as they would be written following the conventions of elementary algebra.

• Let and . Then

  Polynomial addition is defined as

  Where

  Polynomial multiplication is defined as

  Where

• More generally, if we have indeterminates , we denote the polynomial ring in these indeterminates as .

  • An important property, for multiple indeterminates, we have that

• (Fraleigh 22.2) is a ring under polynomial addition and multiplication. If is commutative then so it . If has a unity , then is also the unity in ring .

• (Fraleigh e22.24 ) If is an Integral Domain then so is

• (Fraleigh 22.4) The Evaluation Homomorphisms for Field Theory ¹ Let be a subfield of field and . Let be the indeterminate. The map defined by

  For is a homomorphism of into .

  Also and maps isomorphically by the identity map : .

  We refer to as an evaluation at

  • Intuition: This generalizes the elementary algebra notion of substituting in the polynomial and evaluating

• Let be a subfield of and Let and be the evaluation homomorphism.

  Let denote

  If , then is a zero of .

• (Fraleigh e26.23) Let be a field and let . The set

  is an Ideal in

• (Fraleigh 27.24) If is a field, every Ideal in is principal

  • Intuition: There are three cases to consider
    • Case : The trivial ideal which is clearly principal because it is equal to
    • Case 2: The ideal over constants (i.e., degree polynomials). Clearly these are all units therefore they are generated by .
    • Case 3: Polynomials of degree . Let be the ideal. Take the minimal degree element in this ideal. By the division algorithm . But also
      By closure of an ideal. But, clearly since is minimal. Therefore, this ideal is — the principal ideal with as the polynomial of minimal degree (up to constant multiples).

• (Fraleigh 27.25) An ideal of is maximal if and only if is irreducible over .

  • Intuition: In the forward direction, if the ideal is maximal, it is also prime (since is a commutative ring with unity). Any factorization of must have factors that are multiples of which cannot be.

    In the reverse direction, if we consider the principal ideal of , any hypothetical ideal containing it must also be a principal ideal. This implies is some multiple of but this is impossible because it is irreducible.

• A polynomial is monic if the coefficient of the highest power of appearing is .

• A Power Product in is an expression

  Where , .

  • An ordering of power products can be defined as follows
    • for all power products .
    • For any two power products and exactly one of
      Holds
    • If and then
    • If then for any power product .

Topics §

• Factorization of Polynomials
• Groebner Basis
• Extension Field and Polynomials

Links §

• Fraleigh
• Abstract Algebra Eisenstein’s Criterion by Michael Penn

Footnotes §

1. This theorem holds for commutative rings as well. ↩