PUMaC 2012 · 加试 · 第 3 题

3. C for the set of complex numbers.

For convenience, let Z be the set of positive integers. We write Z [ X ] for the set of polynomials in X with coefficients in Z , and more generally, Z [ X , . . . , X ] for 1 n the set of polynomials in X , . . . , X with integer coefficients. Similar notation 1 n

is used for Q and C . We write n ∈ Z to denote that n is a member of Z . Recall that, like real numbers, complex numbers have a notion of absolute √ 2 2 value: by definition, | a + bi | = a + b . For all z , z ∈ C , we have | z z | = 1 2 1 2 | z || z | and | z + z | ≤ | z | + | z | , just like for real numbers. 1 2 1 2 1 2 2.1 Definition. Let f be a polynomial in X , . . . , X whose coefficients are 1 n not all zero. If f consists of one term, then the degree of f , written deg f , is the sum of the exponents of the X ’s. In general, deg f is the maximum degree j among all terms of f . d For instance, if f ( X ) = a + a X + . . . + a X and a 6 = 0 , then deg f = d 0 1 d d as usual. Henceforth, we always assume a 6 = 0 when writing a polynomial in d d this form. We say that a X is the leading term of f , and a is the leading d d coefficient . Note that the degree of the 0 polynomial is undefined . 2.2 Definition. A polynomial f ( x , x , . . . , x ) is homogeneous if (and only if) 1 2 n each of its terms, individually, has the same degree as the others. Equivalently, f is homogeneous if deg f f ( λx , λx , . . . , λx ) = λ f ( x , x , . . . , x ) 1 2 n 1 2 n for any λ 6 = 0 . 2.3 Definition. A number is algebraic if it is the root of a nonzero polynomial in Q [ X ] . All numbers that are not algebraic are transcendental . We write Q for the set of algebraic numbers. 2.1 Remark. Important! Q is a subset of C . More precisely, every polynomial in Q [ X ] splits completely into linear factors in C [ X ] . The multiplicity of α as a root of a polynomial is the exponent to which ( X − α ) appears in the linear factorization of that polynomial in C [ X ] . 2.4 Definition. Let α ∈ Q . The degree of α , written deg α , is the minimum degree among degrees of all nonzero polynomials in Q [ X ] that have α as a root. 2.5 Definition. A polynomial in one variable is monic if its leading coefficient is 1 . A number is an algebraic integer if it is the root of a monic polynomial in Z [ X ] . 2 √ Here’s an example. Let α = 3 . Note that α is a root of the polynomial 2 f ( X ) = X − 3 . Since α is not the root of a linear polynomial with rational coefficients, α is an algebraic number of degree 2 . Since f is monic and has integer coefficients, α is actually an algebraic integer . d 2.6 Definition. Let f = a + a X + . . . + a X ∈ C [ X ] . Define 0 1 d ‖ f ‖ = max {| a | , . . . , | a |} 0 d 2.2 Remark. If S is a set, then max S is the maximum among the elements of S . The following problems will not only develop our intuition for ‖ f ‖ , but will also be useful later in the test. 2.1 (4 points) Let f, g ∈ C [ X ] such that f 6 = 0 , and let α, β ∈ C .