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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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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47472cc3-819f-4272-92fe-e39219493627%2F34f95f3a-4849-4d35-8b12-ae7d91217e7e%2Ffgvyu83_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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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