Polynomials rings

Question 1

Polynomials rings

Answer 1

Let R be a ring. A polynomial f(x) with coefficients in T is an infinite formal sum

Sum aix^i=a0+a1x+…+anx^n+…

Question 2

R[x]

Answer 2

The set R[x] of all polynomials in an indeterminate x with coefficients in a ring R is a ring under polynomial addition and multiplication. If R is commutative, then so is R[x], and if R has unity 1 ̸= 0, then 1 is also unity for R[x].

Question 3

The Evaluation Homomorphisms

Answer 3

The Evaluation Homomorphisms for Field Theory) Let F be a subfield of a field E, let α be any element of E, and let x be an indeterminate. The map φα : F[x] → E defined by
φ a(a0 +a1x+···+an x^n􏱝=a +a α+···+a αn

Question 4

Let φ: R1 → R2 be a ring homomorphism. Show that there is a unique ring homomorphism ψ: R1[x] → R2[x] such that ψ(a)=φ(a) for any a∈R1 and ψ(x)=x.

Answer 4

Suppose ψ: the homomorphism of R1→R2 be shown by
ψ(a0+a1+a2+a3)=ψ(a0)+ψ(a1x)+ψ(anx^n)
φ is a homomorphism φ(aj+bj)=φ(aj)+φ(bj)
ψ is a ring homomorphism. ψ is unique since the homomorphism of R1 and R2 is directly determined by the value of R1 and X