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let k be a field. verify that the set k[x] of all polynomials with coefficients in k, under the usual addition and multiplication operations, forms a ring. in particular, you must verify that there are additive and mul- tiplicative identities, that addition and multiplication are commutative and associative, that every polynomial has an additive inverse, and that the dis- tributive law holds. do not write out detailed proofs of all this; just specify what the additive identity is, what the additive inverse of a polynomial is, and what the multiplicative identity is.

Answer :

Let K be a field in which (to demonstrate K[n] of all bolynomial coefficients in K below the typical addition and multiplication operations forms a ring]

Since the addition of two polynomials is once more, (i) (To show = (K(n), +) is abelian grouk) is closed under addition (of binary operation).

polynomial a Associativity under addition, assuming that Pa 2, 2 exists in k[u].

K[n] is closed when added, and qinis both (n+bente -+ n, mand p are all different) or nemcp: K + + + (An+ba) +- + = (90+bot (o) +(9, +b, +G) K € - - + p(x+2(m) + 2(x) = ((Qutbe) +(,+bi) K + + + --+ Cpn3 = (bini)+ (qini-eron) K[n] is an addition associativite.

If H 2(4)=0 is additive identity, then clearly pinto-b =10 belynomial is additive identity of KO OR let pin's Po+?не-томни екси.

The inverse element of K[n] is additive inverse- let b(m + 2(120)

To learn more about bolynomial coefficients here

https://brainly.com/question/20630027

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