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Factorization 1. Is z xx] a PID? A UFD? Consider in Z [x] all polynomials with even free term. Show that they form an ideal. Is this ideal: principal? Prime? Maximal? 2. In Z2[x] (polynomials with coefficients in integers mod 2), find all irreducible polynomials of degree <5. Prove that for every non-constant polynomial f(x) in 22[x].f(x2) is reducible. 3. Show that a polynomial reducible in Z[x] remains reducible in Zp [x] (where ois prime). Derive that in Q[x], there are infinitely many monic irreducible polynomials of degree 4. 4. Given m, n>0, find the greatest common divisor of and xT-1. 5. Show that complex numbers of the form a + b-3 form a subring (call it R) in C, and that R is an integral domain. Show that the equality 2² - +V-3)(1 V-3) provides two distinct factorizations of the same element of R into irreducibles (so that R is not a UFD). Show that in R, 2 is irreducible but not prime. (Hint: To establish irreducibility of certain elements in R, it may be convenient to consider the map R--> Z which sends a + D-V to (a + bv-3)(a+bv-3) - a² + 3b² and to notice that this mapis multiplicative.)

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Z[x] is not PID because if we consider an ideal <2,x> which is principal then <2,x> = <h(x)> for some h(x) belongs to Z[x] now we consider p(x) belongs to Z[x] such that p(x).h(x) = 2. Since multiplication of two polynomial is constant therefore deg (h(x)) = 0 . and then h(x) = +1 or -1 . therefore <h(x)> = Z[x] = <2,x>
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