At the end of last lecture we stated a theorem giving an easy characterization of subfields of a given field We begin by proving this result.

Theorem 18.Suppose is a field and If satisfies the following 5 conditions, then s a subfield of

- is closed under addition.
- is closed under multiplication.
- whenever
- whenever and
- has at least two elements.

**Proof.** Notice that If (and there is some such by condition 5) then by 3, and therefore by 1.

Similarly, First, there is some nonzero element in since has at least two elements by 5, and only one of them at the most can be zero. Let Then by 4, and therefore by 2.

Once we have that the verification of the field axioms (see Definition 1 in lecture 4.1) is straightforward since and is a field.

**Examples. 1. ** so is a subfield of and and is a subfield of

**2.** is a subfield of both the field of 4 elements and the field of 8 elements. However, the field of 4 elements is not a subfield of the field of 8 elements. This is because any nonzero element of the field of 4 elements satisfies while any nonzero element of the field of 8 elements satisfies This means that we would have at least one such that but contradiction.

**3. ** is a subfield of To see this, one verifies conditions 1–5 of Theorem 18. Conditions 1–3 and 5 are straightforward. To check condition 4, notice that if and say with then it cannot be that both and are zero. Then because This is because implies that since otherwise from we also have and we cannot have both being zero simultaneously since But then implies that which we know is not the case.

But then and condition 4 follows.

**Definition 19. **If is a subfield of and the smallest subfield of that contains and has among its elements is denoted by

Next lecture we will show that there is indeed such a smallest subfield. Notice that in Example 3 above, we in fact have This is because if is a field, and then since must be closed under addition and multiplication. But, since is already a field, there can be no smaller subfield of that contains and has as an element.

**4.** Similarly, for any integer (in fact, any rational) Here, we see and as being in rather than in case Note that if is already the square of a rational, then

**5.** Again, we use Theorem 18, and only condition 4 requires special care. Suppose that and that We claim that This is because of the following identity, closely related to the inequality between the arithmetic and the geometric means, see this post for further discussion:

Notice that is the determinant of the matrix so the system of equations has a unique solution and must be rational since they are obtained from by means of the elementary operations (). But this means that or

**Definition 20. **A number is **transcendental** iff it is not a root of any polynomial with rational coefficients.

**6.** The number is known to be transcendental. This is a deep theorem of Ferdinand von Lindemann from 1882. This means that we have no “concrete” description of as in the examples above, since (for example) if for some rational then would be a root of the cubic polynomial On the other hand, it turns out that one describe in easy terms: are polynomials with rational coefficients,

[…] Last lecture we characterized subfields and used the characterization to provide many new examples of fields. Now we start to explore systematically which subfields of the complex numbers are suitable to study the question of which polynomial equations can be solved. […]

[…] knew that Corollary 8 holds in a few particular cases, for example for for See also lecture 4.5. The argument above is much more general, and reduces significantly the amount of computations […]