Let K be a field. Observe that the polynomials in K[x] that are not zero and...

Let K be a field. Observe that the polynomials in K[x] that are not zero and not units are precisely the polynomials of positive degree.

Expert Solution

The question says that only units in polynomial ring K[x] are units in K. Since only polynomials of 0 degree are units in K[x].

Colby Messinger answered 1 year ago

Let F be a field. (a) Prove that the polynomials a(x, y) = x^2 − y^2,...

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Let Poly3(x) = polynomials in x of degree at most 2. They form a 3- dimensional...

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In this exercise, we will prove the Division Algorithm for polynomials. Let R[x] be the ring...

In this exercise, we will prove the Division Algorithm for polynomials. Let R[x] be the ring of polynomials with real coefficients. For the purposes of this exercise, extend the definition of degree by deg(0) = −1. The statement to be proved is: Let f(x),g(x) ∈ R[x][x] be polynomials with g(x) ? 0. Then there exist unique polynomials q(x) and r(x) such that f (x) = g(x)q(x) + r(x) and deg(r(x)) < deg(g(x)). Fix general f (x) and g(x). (a) Let...

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Let f(x) = sin(x) for -pi < x < pi and be ZERO outside this interval...

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