A generalisation of the fundamental theorem of algebra

Question

We already know the following theorem by d'Alembert and Gauss, often called fundamental theorem of algebra.

Theorem

Let $P$ be in $\mathbb C[X]$ of degree $1$ or greater. There exists $\alpha\in \mathbb C$ such that $P(\alpha)=0$.

Can we give the following generalisation for polynomials with several variables?

Generalisation

Let $n$ be in $\mathbb N^*$ and $P$ be in $\mathbb C[X_1,\ldots,X_n]$ of degree $1$ or greater. There exists $\alpha\in \mathbb C^n$ such that $P(\alpha)=0$.

Any references, hints or solutions would be greatly appreciated.

If you require that $P$ is not a constant this is true. See weak Nullstellensatz — user2520938, Dec 19 '16 at 16:06
@user2520938 Thank you ! That is what I was hoping, I will try to understand weak Nullstellensatz. — E. Joseph, Dec 19 '16 at 16:09

score 3 · Accepted Answer · answered Dec 19 '16 at 16:41

Suppose, the claim is true for some $n\ge 1$. A polynomial with variables $x_1,\cdots x_{n+1}$ can be considered to be a polynomial with variables $x_1,\cdots, x_n$ and coefficients depending on $x_{n+1}$.

Consider any coefficient belonging to a term containing at least one of the variables $x_1,\cdots , x_n$. (If there is no such coefficient, the polynomial only depends on $x_{n+1}$, leading to the case $n=1$)

The coefficient is either constant or a polynomial of $x_{n+1}$. So, there must be a choice for $x_{n+1}$, such that the given coefficient is not $0$.

Therefor, we can choose an $x_{n+1}$, such that the resulting polynomial in $x_1,\cdots , x_n$ is not constant. By induction hypothesis, this polynomial has a root, hence the given polynomial has a root as well.

A generalisation of the fundamental theorem of algebra

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