Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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- 26 2. Let R = {a + bv2 | a, b e Z} and let R' consist of all 2 by 2 matrices of the form On the last homework, you showed R was a ring (in fact it is a subring of R). Show that R is a subring of Mat2(Z). Then show that p: R R' defined by a 26 p(a + bv2) a is an isomorphism.arrow_forward15. (a) Show that S= {(a,a) | a e Z} is a subring of ZxZ. (Use the definition of addition and multiplication of direct products.) (b) Determine if T = {(a,-a) | ae Z} is a subring of ZxZ. 16. Let R be a commutative ring with unity and letarrow_forward2. Assume that the set S I, y is a ring with respect to matrix addition and multiplication. D-r is a ring homomorphism from (a) Verify that the mapping e : S Z defined by e R to Z. (b) Find ker 0. (c) Find S/kere. (d) Find an isomorphism from S/kere to Z.arrow_forward
- Determine if each of the following maps is a ring homomorphism. (a) f : R → R defined by f(x) %3D -x 1 -2 = (6 1) a ( (b) g : M2(IR) → M2(R) defined 1arrow_forward(19) Let om : Z → Zm be the ring homomorphism defined by om (a) remainder of division of a by m. (a) Show that m : Z[x] → Zm[x] defined by om (anx + + a₁x + ao) = om(an)x² + is a ring homomorphism onto Zm[x]. + om (a₁)x+om (ao) = (b) Show that if f(x) = Z[x] and ¯m (f(x)) = Zm[x] both have degree n and σm(f(x)) does not factor in Zm[x] into two polynomials of degree less than n, then f(x) is irreducible Q[x]. (c) Use the previous part to show that x³ + 17x + 36 is irreducible in Q[x].arrow_forwardNumber 6: show at K is splitting over Q.arrow_forward
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