TRUE OR FALSE Review, give explanation The notation F always denotes a field. 1)
ID: 3210025 • Letter: T
Question
TRUE OR FALSE
Review, give explanation
The notation F always denotes a field.
1). A polynomial over a finite field is irreducible if and only if it has no roots.
2). If F is a finite field and p(x) is a polynomial of degree at least one, then F[x]/(p(x)) has finitely many elements.
3). If F is an infinite field and p(x) is a polynomial of degree at least one, then F[x]/(p(x)) has infinitely many elements.
4). If F is a finite field, there are only finitely many irreducible polynomials over F.
5). If R is a domain, so is R[x].
6). If F is a field, so is F[x].
7). If F is a field, so is F[x]/(p(x)), where p(x) is some polynomial over F.
8). If p(x) Z2[x] has degree two, then Z2[x]/(p(x)) has four elements.
9). Every ring of the form Z2[x]/(p(x)), where p(x) has degree two, is isomorphic to every other such.
10). If p(x) F [x] is irreducible, then (p(x), f (x)) is either 1 or p(x).
11). F[x]/(xa)=F foranyaF.
12). The ring Z5[x]/(x5 + x4 + 1) has 55 elements,
13). The ideal generated by polynomials {x2(x 1)2, (x2 + 1)(x + 2)} in R[x] is the whole ring R[x].
14). The class [x3] is a unit in R[x]/((x + 1)2(x + 2)3(x + 6)5).
15). The polynomial xp x in Zp[x] factors into p polynomials of degree 1.
16). There is a bijection between Zp[x]/(g(x)) and Zp[x]/(h(x)) if and only if the polynomials g and h have the same degree.
17). The rings Zp[x]/(g(x)) and Zp[x]/(h(x)) are isomorphic if and only if the polynomials g and h have the same degree.
18). The rings Z5[x] and Z7[x] are isomorphic.
19). The polynomial x2 + bx + c in R[x] is irreducible if and only if b2 4c is negative.
20). If two (finite) rings are isomorphic, they must have the same number of elements.
21). If ab + cd = 1 for a, b, c, d F[x], then [a] is a unit in F[x]/(d).
22). If R is ring, and f and g are polynomials of degree d and e respectively in R[x], then the degree fg is d + e.
23). In Z17[x]/((x 1)7), the class of the polynomial [x17] is nilpotent.
24). For every positive integer n, there exists a ring of the form F[X]/(p(x)) which has n elements.
25). The map R R[x] sending an element in R to the constant polynomial is a homomorphism.
26). The map Z2 Z5[x] sending [0]_2 to the constant polynomial [0]_5 in Z_5[x] and [1]_2 to the constant polynomial [1]_5 in Z_5[x] is an injective homomorphism.
27). The map F[X] F sending an polynomial f(x) to its constant term is a homomor- phism.
28). The ring Z_5/(x^3 + x + 1) is a field.
29). Fix a F. The map F[X] F sending an polynomial f(x) to f(a) is a homomorphism.
30.) The rings Z_3[x]/x^3 and Z_3[x]/(x^3 + x^2 + x + 1) are isomorphic.
31). The remainder when f(x) F[x] where F is a field, when f is divided by (x a) is f (a).
32). Let f(x) = a(x) b(x) be the essentially unique factorization of a polynomial f Z_2[x] into irreducibles. Then Z_3[x]/(f) has exactly two zero divisors.
33). If p(x) is irreducible and p(x) divides f(x) g(x), then p(x) divides either f or g.
34). If f is monic, then there is only one factorization of f into monic irreducible polynomials, up to reordering the factors.
35). If f and g have no common factors, then [g] is a unit in F[x]/(f). 36). The ring Zn[x] is a domain.
37). The ring Z_p[x] is a domain if p is prime.
38). The ring Z_p[x] is a field if p is prime.
39). If p(x) F[x] is irreducible, where F is a field, and [g(x)] [h(x)] = 0 in F[x]/(p(x)), then p appears in a factorization of either f or g into irreducibles.
40). If f(x) and g(x) are polynomials over R, then (f,g) = 1 if and only if f and g have no common roots.
41). If F F is an inclusion of fields, then F [x] F [x] is an inclusion of rings.
42). Let notation be as in 40). The polynomial f(x) F[x] is irreducible if and only if it is irreducible as a an element of F[x].
43). In R[x]/((x^2 + 3)(x^2 7)), the element [x^3 3x] is a unit.
44). A polynomial in F[x] is irreducible if and only if it has no roots.
45). If f and g are polynomials in F [x] with no common factors (other than constants), then the equation [f]T = [h] can be solved uniquely for T in the ring F[x]/(g).
46). If f and g are polynomials in F [x] with no common factors (other than constants), then (f, g) = 1.
47). If f and g differ by a unit in F[x], then (f,g) = 1.
48). In Q[x]/(x^2 3), we have [x]^2 = 3.
49). The ring Q[x]/(x^2 5) is isomorphic to Q(5).
50). The ring R[x]/(x^2 + 1) is isomorphic to C.
51). If uf + vg = 4 in the ring Q[x], then [f] is a unit in Q[x]/(g).
52). The ring R[x] is a domain for any commutative ring R with identity. 53). In R[x], the product of two monic polynomials can be zero.
Explanation / Answer
1)A polynomial over a finite field is irreducible if and only if it has no roots. this is flase because if polynomial is in the finite field interval then the polynomial comsists of a atleast on root
2) If F is a finite field and p(x) is a polynomial of degree at least one, then F[x]/(p(x)) has finitely many elements.it is true because the finite field means that they has finite number and polynomial is also finite then the division of the finite and finite means finite elements
3). If F is an infinite field and p(x) is a polynomial of degree at least one, then F[x]/(p(x)) has infinitely many elements it is also true similar to the above one imfinite divide finite in infinitely
4) If F is a finite field, there are only finitely many irreducible polynomials over F. it is also ture similar to the above 1st eample
5) if r is a domain function R[X] is also becomes a domain of the X
6) If F is a field, so is F[x] is a field of X.
7). If F is a field, so is F[x]/(p(x)), where p(x) is some polynomial over F. it is also true because the of the explanation of 2
8) If p(x) Z2[x] has degree two, then Z2[x]/(p(x)) has four elements it is flase because we can estimate the without knowing the polynomial of degree
9)
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