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only if {0} is a prime ideal? (m) If I = (x³-2x) is an ideal of Z[x], then what is a polynomial p(x) = Z[x] that has as low a degree as possible and such that x6 + x5 + I = p(x) + I? (n) Let F3 (Z/3Z, +,), R = F3[x], and I = (x3 + x + 2). Is R/I a field with 27 elements? = (o) Let F3 = (Z/3Z, +,), R = F3[x], and I = (x3 + 2x + 1). Is R/I a field with 27 elements?

only if {0} is a prime ideal? (m) If I = (x³-2x) is an ideal of Z[x], then what is a polynomial p(x) = Z[x] that has as low a degree as possible and such that x6 + x5 + I = p(x) + I? (n) Let F3 (Z/3Z, +,), R = F3[x], and I = (x3 + x + 2). Is R/I a field with 27 elements? = (o) Let F3 = (Z/3Z, +,), R = F3[x], and I = (x3 + 2x + 1). Is R/I a field with 27 elements?

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.3: Factorization In F [x]
Problem 13E
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only if {0} is a prime ideal? (m) If I = (x³-2x) is an ideal of Z[x], then what is a polynomial p(x) = Z[x] that has as low a degree as possible and such that x6 + x5 + I = p(x) + I? (n) Let F3 (Z/3Z, +,), R = F3[x], and I = (x3 + x + 2). Is R/I a field with 27 elements? = (o) Let F3 = (Z/3Z, +,), R = F3[x], and I = (x3 + 2x + 1). Is R/I a field with 27 elements?
Transcribed Image Text:only if {0} is a prime ideal? (m) If I = (x³-2x) is an ideal of Z[x], then what is a polynomial p(x) = Z[x] that has as low a degree as possible and such that x6 + x5 + I = p(x) + I? (n) Let F3 (Z/3Z, +,), R = F3[x], and I = (x3 + x + 2). Is R/I a field with 27 elements? = (o) Let F3 = (Z/3Z, +,), R = F3[x], and I = (x3 + 2x + 1). Is R/I a field with 27 elements?
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