Abstract Algebra #36 only 220Part IVRings and Fields36. (Remainder Theorem) Let f(x) e FIx] where F is a field and let a G F Show that the remainder r(x) whenf(x) is divided by x –a, in accordance with the division algorithm, is f(@).37. Let om : Z→ Zm be the natural homomorphism given by g...(a) = (the remainder of a when divided by m)for a e Z.a. Show that Om : Z[x] → Zm[x] given byOm (ao + ajx +· ··+ anx") = 0m(ao) + om (a1)x +...+ Om(an)x"..is a homomorphism of Z[x] onto Zm[x].b. Show that if f(x) e Z[x] and Tm(f(x)) both have degree n and om( f (x)) does not factor in Zm[x] into twopolynomials of degree less than n, then f(x) is irreducible in Q[x].c. Use part (b) to show that x + 17x + 36 is irreducible in Q[x]. [Hint: Try a prime value of m that simplifiesthe coefficients.]SECTION 24 ÎNONCOMMUTATIVE EXAMPLESfons MThus far, the only example we have presented of a ring that in not commutative is thering M,(F) of all n x n matrices with entries in a fieldnothing with noncommutative rings and strictly skew firimportant noncommutative rings occurring very n~examples of such rings.FRCbe doing almostat there arewe givRings of Endom

Here we have given that fx∈Fx, where F is a field and α∈F. it is required to show that the remainder…

30. (Remainder Theorem) Let f(x) e F[x] where F is a field, and let a E F. Show that the remainder r(x) when f(x) is divided by x a, in accordance with the division algorithm, is f(a). 7. Let om : Z→ Z. be the naturol ->

30. (Remainder Theorem) Let f(x) e F[x] where F is a field, and let a E F. Show that the remainder r(x) when f(x) is divided by x a, in accordance with the division algorithm, is f(a). 7. Let om : Z→ Z. be the naturol ->

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Abstract Algebra #36 only

220
Part IV
Rings and Fields
36. (Remainder Theorem) Let f(x) e FIx] where F is a field and let a G F Show that the remainder r(x) when
f(x) is divided by x –
a, in accordance with the division algorithm, is f(@).
37. Let om : Z→ Zm be the natural homomorphism given by g...(a) = (the remainder of a when divided by m)
for a e Z.
a. Show that Om : Z[x] → Zm[x] given by
Om (ao + ajx +· ··
+ anx") = 0m(ao) + om (a1)x +...+ Om(an)x"
..
is a homomorphism of Z[x] onto Zm[x].
b. Show that if f(x) e Z[x] and Tm(f(x)) both have degree n and om( f (x)) does not factor in Zm[x] into two
polynomials of degree less than n, then f(x) is irreducible in Q[x].
c. Use part (b) to show that x + 17x + 36 is irreducible in Q[x]. [Hint: Try a prime value of m that simplifies
the coefficients.]
SECTION 24 ÎNONCOMMUTATIVE EXAMPLES
fons M
Thus far, the only example we have presented of a ring that in not commutative is the
ring M,(F) of all n x n matrices with entries in a field
nothing with noncommutative rings and strictly skew fir
important noncommutative rings occurring very n~
examples of such rings.
FRC
be doing almost
at there are
we giv
Rings of Endom

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