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Evaluation at a point has kernel (x_1-a_1, ..., x_n-a_n)

Statement

Let $k$ be a field and let $a = (a_{1}, \dots, a_{n}) \in k^{n}$ . The evaluation map

{ev}_{a} : k [x_{1}, \dots, x_{n}] \to k, f \mapsto f (a),

has kernel $(x_{1} - a_{1}, \dots, x_{n} - a_{n})$ . In particular, this ideal is maximal.

Facts & Assumptions

Given: A field $k$ , a point $a = (a_{1}, \dots, a_{n}) \in k^{n}$ , and the evaluation map at $a$ .

[L1]

Polynomial evaluation at a point is well defined (Evaluation and roots of a polynomial in a commutative target ring).

[L2]

Division by a monic polynomial produces quotient and remainder (Division by a monic polynomial over a commutative ring).

[L3]

A quotient ring is a field exactly when the ideal is maximal ( $R / M$ is a field if and only if $M$ is a maximal ideal).

Proof

technique · induction on the number of variables

1.1

Base case $n = 1$ : divide $f (x_{1})$ by the monic polynomial $x_{1} - a_{1}$ . By [L2], $f (x_{1}) = (x_{1} - a_{1}) q (x_{1}) + r$ with $r \in k$ . Evaluating at $x_{1} = a_{1}$ gives $r = f (a_{1})$ by [L1], so $f - f (a_{1}) \in (x_{1} - a_{1})$ . Therefore the kernel of ${ev}_{a_{1}}$ is $(x_{1} - a_{1})$ .

L1L2base

1.2

Inductive hypothesis: assume the statement for $n - 1$ variables.

2.1

Write $f$ as a polynomial in $x_{n}$ with coefficients in $k [x_{1}, \dots, x_{n - 1}]$ and divide by the monic polynomial $x_{n} - a_{n}$ : $f = (x_{n} - a_{n}) q + r,$ where $r \in k [x_{1}, \dots, x_{n - 1}]$ . Evaluating at $a$ gives $f (a) = r (a_{1}, \dots, a_{n - 1})$ . By the induction hypothesis, $r - r (a_{1}, \dots, a_{n - 1}) \in (x_{1} - a_{1}, \dots, x_{n - 1} - a_{n - 1})$ , and hence $f - f (a) \in (x_{1} - a_{1}, \dots, x_{n} - a_{n}) .$

L1L2step 1.2

3.1

Step 2.1 shows that every polynomial differs from its value at $a$ by an element of $(x_{1} - a_{1}, \dots, x_{n} - a_{n})$ , so the quotient by that ideal is isomorphic to $k$ . By [L3] the ideal is maximal, and its kernel description is the one established above.

L3step 1.1step 2.1discharge-induction∎

Depends on

Used by

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Sources

J. S. Milne, A Primer of Commutative Algebra, v4.03, Corollary 13.9 (standard reference, not scraped)