MATH2401 Abstract Algebra (4): Field

This article studies how fields grow when elements and roots are adjoined, with particular attention to minimal polynomials, extension degrees, splitting fields, finite fields, Frobenius, and primitive elements.

Collections of MATH2401 Abstract Algebra notes:

1. Group;
2. Ring;
3. Module;
4. Field (this article);
5. Action;
6. Galois Theory.

Extensions and generated fields

Definition. If are fields, is a field extension. Its degree is , the dimension of as a vector space over . The extension is finite if this dimension is finite and algebraic if every element of is algebraic over .

Definition. The prime subfield of a field is its smallest subfield, equivalently the intersection of all subfields of .

Proposition. Every field contains a unique prime subfield. It is isomorphic to when and to when .

For , distinguish

from the smallest subfield , whose elements are rational expressions with nonzero denominator.

Definition. An extension is simple if for some . Such an element is called a primitive element of the extension.

Algebraic and transcendental elements

Definition. An element is algebraic over if for some nonzero ; otherwise it is transcendental. The evaluation homomorphism is

If is algebraic, the kernel is a nonzero principal ideal; its monic generator is the minimal polynomial of over . Every other generator differs by a nonzero scalar, so the monic normalization ensures uniqueness.

Proposition. Minimal-polynomial properties. If is algebraic over , then is irreducible, exactly when , and

If is transcendental, then and .

Proof. Write . First, if with both factors nonconstant, then forces a polynomial of smaller degree to vanish at . Thus is irreducible and is maximal. Second,

is therefore a field. Third, by the definition of the generated field, . Moreover,

If is transcendental, evaluation is injective, giving and, on fraction fields, . ◻

Corollary. If , then is an -basis of , and .

Proof. For every , polynomial division gives

Hence , giving spanning. If , minimality forces , giving linear independence. ◻

Proposition. If are algebraic over , then

and this is a finite extension of . Conversely, if the finitely generated -algebra on the left is a field, every is algebraic over .

Proof. Adjoin the algebraic elements one at a time; each step is finite and is already a field, so the composite is finite and equals the generated field. The converse is Zariski’s lemma: a field finitely generated as an algebra over a field is finite algebraic over it. ◻

Proposition. For algebraic elements and over , there is an -isomorphism sending to if and only if ; when it exists, it is unique.

Proof. An -isomorphism carrying to carries every polynomial relation of one to the other, so the monic minimal polynomials agree. Conversely, identify both fields with ; the resulting isomorphism sends the class of , hence , to . This requirement determines the image of every rational expression, proving uniqueness. ◻

Theorem. Eisenstein’s criterion. If and a prime satisfies

then is irreducible over .

Degrees and towers

Proposition. Every finite extension is algebraic. A simple extension has

when is algebraic and has infinite degree when is transcendental.

Proof. In a finite extension, are linearly dependent, so every is algebraic. The algebraic degree formula follows from the simple-extension degree corollary above; if is transcendental, its powers are linearly independent, so . ◻

Theorem. Tower law. For fields ,

with cardinal multiplication in general. In particular, if both degrees on the right are finite their product is the ordinary integer product; if either is infinite, then is infinite.

Proof. Let be an -basis of and a -basis of . For ,

Moreover,

Thus is an -basis of , and . ◻

Definition. The relative algebraic closure of in is

This is a subfield of , and is algebraic. Indeed, if are algebraic, the finite extension contains , , and when ; every element of a finite extension is algebraic.

Adjoining roots and splitting fields

Theorem. Kronecker. Every nonconstant has a root in some field extension of .

Theorem. Existence and uniqueness of splitting fields. Every nonconstant has a splitting field over , unique up to an -isomorphism.

Proof outline. Existence follows by adjoining roots successively. For uniqueness, if and are roots in two extensions of the same irreducible , then

Starting with this isomorphism and adjoining the remaining roots one at a time extends it inductively. Since each splitting field is generated by all the roots, the final -embedding is surjective and hence an -isomorphism. ◻

Proposition. Multiple-root criterion. An element is a multiple root of if and only if . An irreducible has a multiple root in a splitting field if and only if . Thus irreducible polynomials over a field of characteristic have distinct roots.

Finite fields and Frobenius

Every finite field has characteristic and is a finite-dimensional vector space over ; dimension gives order .

Theorem. Classification of finite fields. For every prime power there is a field of order , unique up to isomorphism. Its multiplicative group is cyclic; a generator is called a primitive element of the finite field.

Proof outline. The roots of in an algebraic closure form a field: Frobenius makes them closed under addition, subtraction, multiplication, and inversion. The derivative is , so there are exactly distinct roots. Any field of that order consists of these roots, proving uniqueness. Cyclicity of the multiplicative group follows from the root-count argument proved in the cyclic groups section. ◻

Proposition. Frobenius. In characteristic , , , is an injective field endomorphism. It is an automorphism when is finite. For it has order and fixed field .

Proof. The binomial coefficients between the endpoints are divisible by , giving ; multiplicativity is clear. A field homomorphism is injective, and on a finite set it is therefore surjective. Fermat’s little theorem gives for every ; since has at most roots, these are precisely the fixed points. Since every obeys , the order divides ; a smaller order would give at most fixed elements, so the order is exactly . ◻

Theorem. Subfields of finite fields. The subfields of are precisely the unique fields for divisors .

Proof. A subfield of order makes a vector space over it, so by the tower law. For , the roots of inside form the required field. They are uniquely determined as the fixed field of the -th power of Frobenius. ◻

Theorem. Factorization of . Over , the polynomial is the product, without repetition, of all monic irreducible polynomials whose degrees divide .

Primitive elements and Bézout identity

Do not confuse a generator of with a primitive element of an extension , meaning an element with .

Theorem. Primitive element theorem. Every finite separable extension is simple. In particular, every finite extension in characteristic , and every finite extension of a finite field, has a primitive element.

Proof outline. If is finite, then is finite and a generator of generates as a field. Suppose is infinite. By induction it is enough to treat . Let and be the distinct conjugates of and (distinct by separability). Choose outside the finite set for which

and put . If and are the minimal polynomials of and , then in the polynomials

have the unique common root . Their monic gcd is therefore , so and then . Thus . Induction handles finitely many generators. ◻

Since is a Euclidean domain, nonzero have a unique monic greatest common divisor , and there are such that . This is Bézout’s identity; the Chinese remainder theorem is the related quotient-ring statement obtained when .