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By Lam T.Y., Magid A.R. (eds.)

ISBN-10: 0821810871

ISBN-13: 9780821810873

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Similarly, the polynomials X 3 + X + 1 ∈ Z2 [X] and X 3 + X 2 + 1 ∈ Z2 [X] are the unique irreducible ones of degree 3 over Z2 , hence Z2 [X]X 3 +X+1 = Z2 [X]X 3 +X 2 +1 = {a + bX + cX 2 ∈ Z2 [X]; a, b, c ∈ Z2 }, having 8 elements, becomes a field in two different ways, being called F8 and F8 , respectively: In F8 we have X 3 = 1+X, X 4 = X +X 2 , X 5 = 1+X +X 2 , X 6 = 1+X 2 , showing that F8 = {0, 1, X, . . , X 6 }, where X 7 = 1 implies that F∗8 = X is cyclic of order 7; in F8 we have X 3 = 1+X 2 , X 4 = 1+X+X 2 , X 5 = 1+X, X 6 = X+X 2 , showing that F8 = {0, 1, X, .

Since pn ∈ R is a prime we may assume that pn | qm , hence since qm ∈ R n−1 m−1 is irreducible we infer pn ∼ qm . Thus we have u · i=1 pi = j=1 qj ∈ R for ∗ some u ∈ R , and we are done by induction. 10) Example: Residue class rings, revisited. We consider again the residue class ring Zn , where n ≥ 2. Then its group of units is given as Z∗n = {a ∈ Zn ; 1 ∈ gcd(a, n)}; note that this yields an alternative proof that Zn is a field if and only if n is a prime: If a ∈ Z∗n then there are k, l ∈ Z such that ka + ln = 1 ∈ Z, implying that 1 ∈ gcd(a, n); if a ∈ Zn such that 1 ∈ gcd(a, n), then there are B´ezout coefficients s, t ∈ Z such that 1 = sa + tn ∈ Z, hence we have sa = 1 ∈ Zn , thus a ∈ Z∗n .

Hence we have r = r and q = q , showing uniqueness. To show existence, we may assume that f = 0 and m := deg(f ) ≥ deg(g) := n. We proceed by induction on m ∈ N0 : Letting f := f − lc(f )lc(g)−1 gX m−n ∈ R[X], the m-th coefficient of f shows that f = 0 or deg(f ) < m. By induction there are q , r ∈ R[X] such that f = q g + r , where r = 0 or deg(r ) < deg(g), hence f = (q g + r ) + lc(f )lc(g)−1 gX m−n = (q + lc(f )lc(g)−1 X m−n )g + r . 13) Polynomial rings over fields. a) Let K be a field. Then we have K ∗ = K[X]∗ = {f ∈ K[X] \ {0}; deg(f ) = 0}, that is the set of non-zero constant polynomials.

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Algebra, K-Theory, Groups, and Education: On the Occasion of Hyman Bass's 65th Birthday by Lam T.Y., Magid A.R. (eds.)


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