Let E be a field and , 6E E be nonzero polynomials. (a) If ab and a, prove that a = db for some nonzero d E E (b) If e gcd(a, b) and E Eis a common divisor of a and b of highest possible degree, prove that i= de for some nonzero d E E

Q: Let ez}. m A = m, n €. m E Z -n (a) Show that A is a subring of the ring Z¿. (b) Show that A is an…

A:

Q: 30. Let E be an extension field of a finite field F, where F has q elements. Let a e E be algebraic…

A: The objective is to prove that, if  is a finite field and  has  elements. If  be an extension field…

Q: 2. Let P(x) = ax² + bx + c be an irreducible polynomial in Q[x). Prove that (a) P(x) has two…

A: Let P(x)=ax2+bx+c be an irreducible polynomial in Q[x] a) To show P(x) has two distinct roots, let…

Q: If f (x) is any polynomial of degree n21 over a field F, then there exists an extension K of F such…

A:

Q: Let F be a field and let f(r) = anr" +an-1x"-1+..+ ao € F[x]. Prove that r - 1 is a factor of f(r)…

A:

Q: Let K be a field of characteristic zero. Show that a polynomial of the form t4+bt² + c is solvable…

A:

Q: Let f(x) and g(x) be irreducible polynomials over a field F and let a and b belong to some extension…

A: Consider fx and gx be irreducible polynomials over a field F and a and b belongs to some extension E…

Q: e) x·y= 0 iff x = 0 or y= 0. ) x<у iff — у < -х.

A:

Q: If f (x) is any polynomial of degree n21 over a field F, then there exists an extension K of F such…

A:

Q: Let K be an extension of a field F. An element a e K is algebraic over F if and only if [F (a) : F]…

A:

Q: Determine the remainder r when f(x) is divided by x - c over the field F for the given f(x), c, and…

A: As per the guidelines we are supposed to answer only three subparts. Kindly repost rest of the…

Q: 2. Find all values of p such that Z„[r]/{x² + 1) is a field.

A: We know that results

Q: Let K be any field, and let a1,...,an be pairwise distinct elements of K (that is, a; + aj for all i…

A:

Q: Let F = {0, 1,2} denote the field of 3 elements. Consider the field K obtained by adjoining the…

A: Given the field of 3 elements F=0,1,2., F=m=3 So, characteristics of field F is 3.

Q: If f (x) is any polynomial of degree n21 over a field F, then there exists an extension K of F such…

A:

Q: Let F be a field and a be a non-zero element in F. If f(x) is reducible over F, then f(x+a)EF[x] is…

A: Use the properties of ring of polynomials to solve this problem.

Q: Let R be a field and let f (x) E R[x] with deg(f (x)) = n > 1. Then, 6. If f (x) has roots over R,…

A:

Q: Suppose that E is the splitting field of some polynomial over a field Fof characteristic 0. If…

A:

Q: et FCK be an algebraic field extension and let a K. Prove that if dimp(F(a)) is odd then F(a) F(a²).

A:

Q: Determine the remainder r whenf(x) is divided by x - c over the field F for the given f(x), c, and…

A: Given, f(x)=x5+x3+x+1 now divide f(x) by (x-1) we can divide f(x) as…

Q: Let f(x) = x² +1 € Z3[x] and let R= Z3[x]/I, where I = (f(x)). (a) Show that R is a field with 9…

A:

Q: FIND the splitting field E of x⁵− 3 over Q and [E : Q]. Justify your answer completely.

A:

Q: a. Show that the field Q(v2, V3) = {a + bv2 + cv3 + dvZV3: a, b, c, d E Q} is a finite extension of…

A: a. The field extension Q(√2, √3), obtained by adjoining √2 and √3 to the field Q of rational…

Q: Find all polynomials p(x) E Z2[x] of degree at most 3 such that Z2[x]/{p(x)) is a field. How many…

A:

Q: Find the degree and a basis for the given field extension: (a) Q(v2, v3, /18) over Q. (b) Q(/2, v6)…

A: find the degree and a basis for the given field extension (a) ℚ2 , 3 , 15 over ℚ let us consider ℚ2…

Q: For a field F find an irreducible polynomial f(x) which generates the same ideal in F[x] as the two…

A: Polynomial which is divisor of each polynomial.

Q: Let FC be the splitting field of x² -2 over and ==e7 Let [F: (=)] = a[F: (√2)] = b then? a) a = b =…

A: Option a is correct i.e. a=b=7.

Q: If f (x) is any polynomial of degree n21 over a field F, then there exists an extension K of F such…

A:

Q: Let F be a field and a(x), b(x) = F[x] not both zero. The greatest common divisor (gcd) of a(x) and…

A:

Q: Suppose f(x) E Z,[r] and f(x) is irreducible over Z,, where p is prime. (a) If deg f(x) = n, prove…

A:

Q: 3. Suppose p is prime and E is a field extension of Zp. Suppose there is a € E which is a zero of rP…

A: Given: p is prime and E is field extension of  Zp. α∈ E  is a root of xp -x+1. (b): To prove: mα,zpx…

Q: Find all values of p such that Z„[x]/(x² + 1) is a field.

A: Given problem is :

Q: For which n listed below does there exist a field extension F ɔ Z/2 of degree n such that the…

A: Follow the procedure given below.

Q: 3. Suppose E is a splitting field of g(x) E F[x] \ F over F. Suppose E = F[a] for some a. Prove that…

A: Let σ∈EmbFE,E.  Given that E=Fα. That is α is algebraic over F and mα,Fx=a0+a1x+...+akxk is the…

Q: be a field and let f(x) = F be of degree n > 1. Let K be an extension field of F a

A:

Q: Let f(x) belong to F[x], where F is a field. Let a be a zero of f(x) ofmultiplicity n, and write…

A:

Q: Let K be the splitting field of – 5 over Q. - • (a) Show that K = Q(V5,i/3) • (b) Explicitly…

A: Hi! For the part (c), we will be needing the information that what all groups we have seen before…

Q: Suppose E is an extension of a field F, and a,b are elements of E. Further, assume a is algebraic…

A: suppose E is an exiension of a field F , a,b are element of E. assume  a is algebraic over F of…

Q: Let F be an infinite field and let f(x) E F[x]. If fſa) = 0 for infinitely many elements a of F,…

A:

Q: a. Show that the field Q(vZ, v3) = {a + bvZ + cv3 + dvzv3:a, b, c, d e Q} is a finite extension of…

A:

Q: Let Q be the field of rational numbers, then show that e(vZ, v3) = Q(vZ + J3).

A:

Q: Let F be a field and let I = {a„x" + a„-1.X"-1 a, + an-1 + · ··+ ao = 0}. ...+ ao I an, an-1, . .. ,…

A: We will test following two things to check if I is an ideal of F[x] (a) For f(x),g(x) in I,…

Q: Let F be a field and let a be a non-zero element in F. If f(ax) is irreducible over F, then…

A:

Q: Let f(x) and g(x) be irreducible polynomials over a field F and let a and b belong to some extension…

A: Let degree of the polynomial fx is n and degree of the polynomial gx is m. Given that fx, gx are…

Q: If F = (M, N, P) and all of the components of this field are polynomials in x, y, and z, then which…

A:

Q: Let D be the set of all real numbers of the form m + n √2 , where m, n ϵ Z. Carry out the…

A:

Q: Find the degree and a basis for the given field extension: (a) Q(/2, v3, v18) over Q. (b) Q(/2, v6)…

A: (a) We have to find the degree and basis of ℚ2,3,18 over ℚ

Q: Let F be a field and let I = {a„x" + a„-|*"-1 + a, + a,-1 + Show that I is an ideal of F[x] and find…

A:

Q: Let F be a field and let f(x) be a polynomial in F[x] that is reducible over F. Then * O is not a…

A: Let F be a field. We say that a non-constant polynomial f(x) is reducible over F or a reducible…

Q: If p(x)∈F[x] and deg p(x) = n, show that the splitting field for p(x)over F has degree at most n!.

A: