the blank: 1. Let F be a field equipped with a Euclidean valuation δ: D\{0} → N∪{0} defined by δ(f)=deg f. If δ(f)=0 then f(x) must be a ____. 2. Let f and g be nonzero polynomials in F[x]. Then the unique greatest common divisor of f and g is the polynomial p that divides both f and g such that p is ____ and of ____. 3. Let F[x] be the ring of polynomials with coefficients over a field F. We say that p is an irreducible polynomial in F[x] iff deg p > 0 and the factors of p must be a unit and its _____.

Q: Let F be a finite field of order q. Let E be the splitting field of Xª – X +1 over

A: Let F be a field of order q Consider the polynomial fX=Xq-X+1∈FX We have to determine the order of E…

Q: Let f(x),g(x) in K[x], a polynomial ring over the field K.  Suppose that g(x)=f(ax+b), where a,b ne…

A: The given data in this question is: f(x) and g(x) are polynomials in K[x]. g(x) = f(ax + b), where…

Q: Let R be the set of all continuous functions. It is known that (R, +,×) is a ring where + and × are…

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Q: for which value of L are the legendre polynomial orthogonal on [ -L, L ]  interval? also find || P2…

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Q: Let F be a field and let R be the integral domain in F[x] generated byx2 and x3. (That is, R is…

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Q: (7) Let F be a field. Prove that (−1) · x = -X.

A: Let F be a field. We have to prove that (-1)·x=-x.

Q: Corollary - Let F be a field. Then F[x] is a unique factorization domain.

A: Given: Theorem 18.3: Every principal ideal domain is a unique factorization domain. Corollary: Let F…