Lecture 09 - Feb 2

Recap: Existence of Algebraic Closure

Theorem

For every field , an algebraic closure exists.

---

Proof (Construction)

Let

Consider the polynomial ring

with one variable for each .

For each , consider the polynomial

Let

be the ideal generated by these elements.

---

Claim

If , then .

So there exist finitely many polynomials and elements such that

Each involves only finitely many variables, so the above equality involves only finitely many variables.

Now suppose is a field extension of containing a root of each .

Substituting , we get

a contradiction.

Hence is a proper ideal.

---

Constructing a Field

Let be a maximal ideal of containing .

Define

Then is a field containing .

Moreover, for each , the image of in is a root of .

Thus contains a root of every non-constant polynomial in .

---

Making it Algebraically Closed

If is already algebraically closed, we are done.

If not, repeat the process: adjoin roots of polynomials over .

If this does not terminate in finitely many steps, take the union

where each is obtained from by adjoining roots of all polynomials over .

---

Why is Algebraically Closed

Let .

Its coefficients lie in some .

By construction, has a root in .

Hence is algebraically closed.

---

Extension of Homomorphisms

Theorem

Let be an algebraic extension, and let be an algebraically closed field.

Let

be a field homomorphism.

Then extends to a field homomorphism

such that

---

Proof

Step 1: Simple Extension

Assume

Define

by applying to coefficients:

Let be the minimal polynomial of over .

Then

Since is algebraically closed, there exists such that

Define

This determines a homomorphism

---

Step 2: General Case (Zorn’s Lemma)

Let

Define a partial order:

Then is nonempty (contains ).

By Zorn’s Lemma, there exists a maximal element .

If , then there exists .

Since is algebraic, is algebraic over .

By Step 1, extends to , contradicting maximality.

Hence

Therefore, extends to all of .

---

Conclusion

Every field homomorphism

extends to a homomorphism

when is algebraic and is algebraically closed.