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Topics covered: polynomits, Euclidean rings, division, reduction modulo p. 3.1. Let A be a commutative associative unital ring, A[x] be the ring of polynomials in one variable x with coefficients in A. Let f=a0+a1x++anxn be a polynomial. Prove the following propositions. A. f is invertible in A[x] if and only if a0 is invertible in A and a1,,an are nilpotent B. f is nilpotent if and only if a0,,an are nilpotent. C. f is a zero divisor if and only if there exists a nou-zero element aA such that af=0. 3.2. A. Show that K[x] ( K a field) is a Euclidean ring 2 with respect to the Euclidean function N(f)=deg(f). B. Similarly, show that the ring of Gaussian integers Z[i]C is Enclidean with respect to the function N(a+bi)=a2+b2. C. Do the same for the so-called Eisenstein numbers which are complex numbers of the form a+b,a,bZ and =e12 the primitive eube root of unity. The Euclidean function is given by f(a+ba)=a2ab+b2. 3.3. A non-zero non-invertible element p of an integral domsin A is called irreducible if it cannot be represented as a product p=ab with a,b non-invertible. Irreducible elements of K(x) are known as irreducible polynomials. Let A be a Euclidean ring. Then if an irreducible element p divides the product a1an then there is a factor ai such that p divides ai (i.e. irreducible elements of A are prime elements). 3.4. Prove that in a Enclidean ring A every non-zero non-invertible element can be factored into a product of irreducible elements, with the decomposition being essentially umique (i.e. unique up to permutation of factors and multiplication by invertible elements). 3.5. (Gauss's lemma) Prove that if a polynomial fZ[x] with integer coefficients is a product f=gh of two polynomials g,hQ[x] with rational coefficients then f is a product of two polynomials g~,h^ proportional to g,h respectively such that g~,h^ are polynomials with integer coefficients. "Recall that an element a of an associative ring is nilpotent if there exists a powitive intecer n such that a=0. 2 It folkws from the previous problem that K[x] is a domain when K is a domaia, in porticular a feld