Algebraic Geometry, Finitely Generated Algebra as a Field

UFD Field Lemma

Let R be a ufd with fraction field F. Assume R has infinitely many primes. This is the case when R = Z, or when R = K[x] for some field K. If K is infinite, each monomial x-c is prime, and if K is finite, there are plenty of irreducible polynomials over K.

Suppose A is a finitely generated R algebra that happens to be a field. Let R embed in A, whence F also embeds in A. Finally, suppose A is algebraic over F.

Let A = R[y₁,y₂,…y_n], where each y_i is a generator of A, and the image of the corresponding indeterminant x_i under the implied ring homomorphism. Each generator y_i satisfies a monic polynomial over F. Let d be a common denominator for all the coefficients of all n polynomials. Since R is a ufd, d is well defined.

Let S be the fraction ring of R by d. Since R is an integral domain it embeds in S. Since A contains F it contains S. Extend our ring homomorphism to S[x₁,x₂,…x_n], mapping onto S[y₁,y₂,…y_n]. The latter ring is simply A.

Note that each generator y_i is integral over S. Assume e is in F-S, so that e is brought in via the introduction of y₁ through y_n. Since there are prime denominators outside of d, there is no trouble finding such an e. Let e = 1/q where q is one of the primes not in d.

Now e lies in the integral extension of S, hence e is integral over S. Let e satisfy a monic polynomial p with coefficients in S. Clear denominators, so that p becomes a polynomnial in R, whose lead coefficient is a power of d, say d^j. Let m be the degree of p. Multiply through by d^j×(m-1). Fold powers of d into e, and d^j/q becomes integral over R. Yet R, a ufd, is integrally closed. This places d^j/q in R, which is impossible, since q does not divide d. Therefore A cannot contain the field F. There is no finitely generated R algebra that is also an algebraic field extension of F.

Finitely Generated Algebra as a Field

Let A be a finitely generated K algebra. If A is a field then A is a finite extension of K.

Use induction on the number of generators. No generators means A = K, and we're done.

One generator means A = K[x] quotiented out by some maximal ideal. Since K[x] is a pid, the generator of the maximal ideal is an irreducible polynomial. The degree of this polynomial equals the dimension of the extension A/K, which is of course finite.

For multiple generators, replace the homomorphism from K[x₁…x_n] onto A with two successive homomorphisms. The first maps x₁ onto its image y₁ in A. The second maps the remaining indeterminants onto their images in A.

Let R = K[y₁] and let F = K(y₁), the fraction field of R. Thus R is a subring of A, and the second homomorphism takes R[x₂…x_n] onto R[y₂…y_n], which is, by assumption, a field. This field contains the fraction field of R, namely F.

Extend the ring R[x₂…x_n] to the ring F[x₂…x_n], and extend the homomorphism accordingly. For starters, F maps to F. The indeterminants map into A as they did before, and that establishes the extended homomorphism.

Now A is finitely generated over F, and by induction, A is a finite extension of F. If F is a finite extension of K we are done.

Suppose y₁ is transcendental. Thus R is a ufd and F is its fraction field. Furthermore, a finitely generated R algebra becomes a field that algebraically extends F. This contradicts the ufd field lemma. Therefore y₁ is algebraic over K.

There is some polynomial p such that p(y₁) = 0. Since R embeds in a field it has no zero divisors. A factor of p would become a zero divisor, hence p is irreducible. The ring extension K[y₁] is a finite field extension over K, and that completes the proof.

Finite Fields

If K is a finite field and the finitely generated K algebra A is also a field then A is a finite field. This follows directly from the previous theorem.

Assume F is a finitely generated field, with no base ring K. In other words, F is the quotient of Z[x₁…x_n].

If F has characteristic 0 it contains Q, the rational numbers. F is a finitely generated Q algebra that is also a field, F is a finite field extension of Q, and F is a finitely generated Z algebra. This contradicts the ufd field lemma. Thus F has characteristic p, and F becomes a finitely generated Z_p algebra. By the above, F is a finite field. A finitely generated ring that is a field is a finite field.

Mapping A into C

Let A be a finitely generated K algebra and let M be a maximal ideal in A. Mod out by M and find a finitely generated K algebra that happens to be a field. As shown above, such a field is a finite extension of K, hence algebraic over K. If C is the closure of K, embed K in C, map M to 0 in C, and carry the quotient field A/M into C, thus giving a ring homomorphism from A into C. The image is L, where L/K is a finite field extension of K inside C.

Conversely, let f be a ring homomorphism from A into C. Since 1 maps to 1, the base field Q or Z_p is fixed. As for the rest of K, combine f with an automorphism on C if necessary, so that f fixes K. Let L be the image of A in C. Now L is integral over K, and L is an integral domain. Since K is a field, L is also a field. The kernel has to be a maximal ideal.

Weak Nullstellensatz

One can use these theorems to prove weak nullstellensatz. (I'll deal with weak and strong nullstellensatz in the next section; this is merely a foretaste.) Let H be a proper ideal in the polynomial ring K adjoin finitely many indeterminants, and let C be the algebraic closure of K. The algebraic set H′ in Cⁿ is nonempty.

Drive H up to a maximal ideal M. Mod out by M and the quotient ring is a field, and a finitely generated K algebra, hence it is a finite extension F/K. For each generator x_i, let z_i be the image of x_i in F. Of course each z_i is algebraic over K, thus building a point z in Cⁿ. If p is a polynomial in our ring, then p, evaluated at z₁ through z_n, is the same as p(x₁…x_n) mapped into F. When p ∈ M, the result is 0. All of M vanishes on z, H vanishes on z, and H′ is nonempty.