Question

Let \(k\) be an infinite field, \(F \in k\left[X_{1}, \ldots, X_{n}\right] .\) Suppose \(F\left(a_{1}, \ldots, a_{n}\right)=0\) for all \(a_{1}, \ldots, a_{n} \in k\). Show that \(F=0 .\) (Hint: Write \(F=\sum F_{i} X_{n}^{i}, F_{i} \in k\left[X_{1}, \ldots, X_{n-1}\right] .\) Use induction on \(n\), and the fact that \(F\left(a_{1}, \ldots, a_{n-1}, X_{n}\right)\) has only a finite number of roots if any \(\left.F_{i}\left(a_{1}, \ldots, a_{n-1}\right) \neq 0 .\right)\)

Short answer

Question: Prove that if a polynomial F in n variables, with coefficients in an infinite field k, evaluates to zero for all possible combinations of its variables, then the polynomial itself is identically zero. Answer: We used induction on the number of variables n to prove that every F_i, corresponding to a single term with X_n raised to power i, is the zero polynomial. We then substituted these zero polynomials back into F and showed that F is identically zero.

Step-by-step solution

Set up induction base case (n = 1)

Let's start with the base case when n = 1. If there's only one variable X_1 in the polynomial F, we have that for each a_1 in k, F(a_1) = 0. Since there are infinitely many elements in k, F must be the zero polynomial by the properties of fields. So, the base case holds.

Set up induction hypothesis

Now, let's proceed with the induction step. Assume that for any polynomial F in n - 1 variables, if it evaluates to zero for all possible values of its variables, then F is the zero polynomial. We'll prove that the same holds for a polynomial in n variables.

Express F as the sum of terms as given in the hint

Based on the hint, express F as the sum of sub-polynomials F_i with only one term with X_n raised to power i: $$F = \sum_{i=0}^{\infty} F_{i}X_n^i$$ where the F_i are polynomials in n - 1 variables, i.e. \(F_{i} \in k\left[X_{1}, \ldots, X_{n-1}\right]\).

Define G

Define a polynomial G in X_n with coefficients in terms of a_1, ..., a_n-1 as: $$G(X_n) = F(a_1, ..., a_{n-1}, X_n)$$ Now for any fixed values of a_1, ..., a_n-1, the polynomial G(X_n) is a polynomial in a single variable X_n and its coefficients are elements of k. By our assumption, G(X_n) = 0 for all possible values of X_n ∈ k.

Use the finite roots property

Since G(X_n) = 0 for infinitely many values of X_n, and it can have at most finite non-zero values, it must be the zero polynomial itself. Therefore, all its coefficients are zero. In terms of F_i, this means: $$F_i(a_1, ..., a_{n-1}) = 0$$ for all i and all possible values of a_1, ..., a_n-1.

Apply induction hypothesis

According to our induction hypothesis, if \(F_i(a_1, ..., a_{n-1}) = 0\) for all possible values of a_1, ..., a_{n-1}, then F_i is itself the zero polynomial. Consequently, we can conclude that all F_i are the zero polynomial.

Conclude that F is identically zero

Since all F_i are the zero polynomial, F reduces to the following expression: $$F = \sum_{i=0}^{\infty} 0X_n^i = 0$$ Therefore, if F vanishes for all possible values of its variables from the infinite field k, it is the zero polynomial. This concludes our induction proof.

Key concepts

Algebraic Curves

Algebraic curves can be thought of as the solutions to polynomial equations in two variables. For example, an equation like \( y^2 = x^3 + ax + b \) defines an elliptic curve. These curves have profound significance in mathematics, particularly in providing insights into the geometric properties of polynomial equations and number theory.

Considering the polynomial zero theorem in the context of algebraic curves, if we have a polynomial in two variables \( F(x, y) \) that vanishes for all points on a curve in an infinite field, this implies that the polynomial must be identically zero. In finite fields, a polynomial can have multiple roots without being the zero polynomial, but the infinite nature of fields in this context is a key factor leading us to the conclusion that \( F \) must be zero everywhere on the curve.

Field Theory

Field theory is a branch of mathematics that studies fields, which are algebraic structures equipped with two operations, addition and multiplication, that roughly follow familiar arithmetic properties. Crucially, in a field, every non-zero element has a multiplicative inverse. Common examples of fields include the set of real numbers \( \mathbb{R} \) and the set of complex numbers \( \mathbb{C} \).

In the context of the polynomial zero theorem, field theory tells us that a non-zero polynomial of degree \( n \) can have at most \( n \) roots in a field. This statement holds true even for infinite fields, such as \( \mathbb{R} \) or \( \mathbb{C} \), and is fundamental to proving that a polynomial that vanishes on an infinite field must indeed be the zero polynomial.

Mathematical Induction

Mathematical induction is a proof technique that allows us to prove an infinite number of statements by linking them together like a chain. To use induction, we first prove a base case. Then, we assume our statement holds for some \( n \) and show that if it holds for \( n \), it must also hold for \( n+1 \).

In our polynomial example, induction helps show that if our statement that the polynomial is zero for \( n-1 \) variables is true, it must also be true for \( n \) variables. Induction is a powerful tool in proving properties of objects defined recursively, as with polynomials over an infinite field.

Infinite Field Properties

In infinite fields, unlike their finite counterparts, polynomials have distinct properties related to roots and factorization. For instance, an important property is that a non-zero polynomial of degree \( n \) cannot have more than \( n \) roots. This property does not always hold in finite fields.

Given the exercise at hand, if a polynomial \( F \) with variables from an infinite field vanishes for an infinite number of inputs, it must be the zero polynomial. This property is unique to infinite fields and is instrumental in the steps of the solution provided as it ensures that a polynomial that evaluates to zero for all elements in an infinite field has to be zero in its entirety.