Basics of fields体たい

A field体たい is a commutative ring可換環かかんかん in which division by nonzero elements is possible. Fields are studied because they make computations in equations, linear algebra, and polynomials stable.

In the integers, the equation $$2x=1$$2x=1 cannot be solved within the integers. In the rational numbers or real numbers, however, it has the solution $$x=1/2$$x=1/2. A field体たい is the structure that expresses this difference.

1Definition of a field体たい

A commutative ring可換環かかんかん $$F$$F is a field体たい if $$0\ne 1$$0\ne1 and every nonzero element $$a\in F$$a\in F has a multiplicative inverse乗法逆元じょうほうぎゃくげん.

It is important to require $$0\ne 1$$0\ne1 and exclude 0 from the inverse condition. If an element $$b$$b satisfied $$0b=1$$0b=1, the basic ring identity $$0b=0$$0b=0 would give $$0=1$$0=1, contradicting $$0\ne 1$$0\ne1. Thus 0 has no multiplicative inverse乗法逆元じょうほうぎゃくげん.

2Basic examples

are fields体たい.

On the other hand, $$\mathbb{Z}$$\mathbb Z is not a field体たい, because the multiplicative inverse乗法逆元じょうほうぎゃくげん $$1/2$$1/2 of 2 is not an integer.

Also, when $$p$$p is prime,

is a field体たい. If the residue class剰余類じょうよるい $$\left[a\right]$$[a] is nonzero, then $$p\nmid a$$p\nmid a. Since $$p$$p is prime, $$\text{[PARSE ERROR: Undefined("Command("gcd")")]}(a,p)=1$$\gcd(a,p)=1. Bezout's identity gives integers $$u,v$$u,v such that

Taking residue classes modulo $$p$$p gives $$\left[a\right]\left[u\right]=\left[1\right]$$[a][u]=[1], so $$\left[u\right]$$[u] is an inverse逆元ぎゃくげん of $$\left[a\right]$$[a].

3Why linear algebra works over fields体たい

In linear algebra, operations that divide by coefficients appear frequently. For example, in Gaussian elimination掃はき出だし法ほう, one divides by a nonzero number to make a pivot主成分しゅせいぶん equal to 1.

This operation is justified because coefficients are elements of a field体たい, and every nonzero element has an inverse.

This section is a preview of applications; results from linear algebra are not used as prerequisites for the later definitions or proofs about fields.

4Theorem: every field体たい is an integral domain整域せいいき

Every field体たい is an integral domain整域せいいき. Proof. Suppose $$ab=0$$ab=0. If $$a=0$$a=0, the conclusion already holds. If $$a\ne 0$$a\ne0, then $$a^{-1}$$a^{-1} exists, and multiplying on the left gives

By associativity, $$\left(a^{-1}a\right)b=b=0$$(a^{-1}a)b=b=0. Thus $$ab=0$$ab=0 implies $$a=0$$a=0 or $$b=0$$b=0, so the field is an integral domain. The inverse is used only after checking $$a\ne 0$$a\ne0; no division by zero occurs.

Conversely, not every integral domain整域せいいき is a field体たい. The ring $$\mathbb{Z}$$\mathbb Z is an integral domain整域せいいき but not a field体たい.

5What changes and what is preserved

Comparing their axioms, a field体たい is an integral domain整域せいいき with the additional requirement that every nonzero element have a multiplicative inverse乗法逆元じょうほうぎゃくげん. Thus a field retains addition, multiplication, the distributive laws, commutativity, and $$0\ne 1$$0\ne1 while allowing division by nonzero elements.

This does not mean that one can merely declare inverses inside the same underlying set of an arbitrary integral domain. For example, $$\mathbb{Z}$$\mathbb Z is an integral domain, but the inverse of its element 2 does not lie in $$\mathbb{Z}$$\mathbb Z, so $$\mathbb{Z}$$\mathbb Z itself is not a field.

6Exercise link

Problems 1–4 and the proof exercise are ready at this point. Complete Problem 5 after the next lecture on finite fields.

7Summary

A field体たい is a commutative ring可換環かかんかん with $$0\ne 1$$0\ne1 in which every nonzero element has a multiplicative inverse乗法逆元じょうほうぎゃくげん. Because division is possible, basic operations in equations and linear algebra are naturally justified.