← MATH 01340 · Modern Algebra I
MATH 01340 · Rings & fields (intro)
Polynomial rings overview
Definition
F[x] is the ring of polynomials with coefficients in F. Degree adds under multiplication for nonzero polys over a domain. Division algorithm holds over fields, yielding gcd theory analogous to integers.
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Irreducible polynomials play the role of primes; unique factorization domains appear.
Evaluation homomorphisms and roots: (x−a) divides f iff f(a)=0.
Quotient F[x]/(p(x)) can build field extensions when p is irreducible.
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Attempt each problem first, then open the worked solution.
1.Divide x³+1 by x+1 in ℚ[x] and interpret the remainder.