4 Subgroups, Subrings and Subfields

Given a group, ring or field, it is natural to ask when a given subset of that algebraic structure obeys the same axioms or inherits the same types of properties as the parent structure.

For example, the rationals form a field contained in the larger field of real numbers. The integers form a ring, but not a field even though they are contained in the field Q. We say that Q is a subfield of R and that Z is a subring of Q. Definition 4.1. Let G be a group wrt the operation and let H be a non-empty subset of G. We say that H is a subgroup of G, and write H < G, if H itself is∗ a group wrt . ∗ We can define analogous substructures corresponding to rings and fields. Definition 4.2. Let R be a ring and let S be a non-empty subset of R. We say that S is a subring of R if S is itself a ring wrt the same operations as R. Definition 4.3. Let F be a field and let K be a non-empty subset of F . We say that K is a subring of F if K is itself a field wrt the same operations as F .

If S is a subring of a ring R we write S < R, and if K is a subfield of a field K we write F

Checking through all axioms of a group/ring/field can be time-consuming. The following results make this much easier. Lemma 4.1. Let G be a group wrt the operation and let H be a non-empty subset of G. Then H is a group wrt iff ∗ ∗ 1. a b H for all a,b H (i.e. is an operation on H), ∗ ∈ ∈ ∗ 2. a−1 H for every a H. ∈ ∈ Proof. If 1 holds then is an operation on H. It must also be assoc on H since if the assoc law holds in G then it must certainly∗ hold for any subset of G. Both 1 and 2 imply that aa−1 = e H for any a H, where e is the identity element of G wrt , so all three axioms of a group are satisfied.∈ The converse∈ is immediate. ∗

4 Example 4.1. The set C4 is a subgroup of C12 wrt complex multiplication. If z C4 then z = 1, so 12 3 4 ∈ z =1 =1 and hence z is also a complex 12th root of unity. It follows that C4 C12. If z, w C4 then 4 4 4 3 4 ⊂3 4 4 3 ∈3 (zw) = z w =1 1=1, so zw C4. Finally, if z C4 then zz = z =1 and (z ) = (z ) =1 =1, so z−1 = z3 C . It· follows that C ∈< C . ∈ ∈ 4 4 12 Example 4.2. The set a b UTGL (Q)= : a,b,c Q,a,c =0 2 0 c ∈ 6    is a subgroup of the group of invertible 2 2 matrices with coefficients in Q, GL2(Q). It is clearly a subset, the product of any pair of invertible upper× triangular matrices is also invertible and upper triangular, and the inverse of any matrix in UTGL2(Q) can be computed as

−1 a b c b =1/ac . 0 c 0 −a     We can apply similar techniques to obtain criteria for determining exactly when a subset of a ring (resp. field) is a subring (resp. subfield). Lemma 4.2. Let R be a ring and let S be a non-empty subset of R. Then S is a subring of R iff 1. a + b S for all a,b S (i.e. + is an operation on S), ∈ ∈

12 2. ab S for all a,b S (i.e. is an operation on S), ∈ ∈ · 3. a S for every a S. − ∈ ∈ Moreover, if R is a field then S is a subfield of R iff 1,2,3 hold and 4. a−1 S for every a S. ∈ ∈ Example 4.3. Consider the set Z(√2) = a + √2b : a,b Z . Clearly Z(√2) R. Let α = a + √2b and β = c + √2d for some integers a,b,c,d. Then{ ∈ } ⊂

α + β = (a + c)+ √2(b + d), αβ = (ac +2bd)+ √2(ad + bc) and α = a + √2( b), − − − so from Lemma 4.2, Z(√2) is a subring of R. If α−1 Z(√2) for any nonzero α in Z(√2) then we’ll have shown that it is in fact a subfield of the field of real numbers.∈ Suppose that α = a + √2b =0. Now α−1 R, since R is a field. We need to check that it does in fact lie in Z(√2). Computing the inverse6 of α, we∈ find that −1 1 1 a √2b a √2b α = = − = 2− 2 . a + √2b a + √2b a √2b! a 2b   − − It is easy to show that a2 =2b2 for any integers a,b, not both zero. However, note that 6 a b , − a2 2b2 a2 2b2 − − are not always integers for any a,b Z. This means that arbitrary α−1 / Z(√2), so Z(√2) is not a subfield of R. On the other hand, we can say∈ that ∈ a b , − Q a2 2b2 a2 2b2 ∈ − − for any pair of integers a,b, not both zero. In fact, applying a similar argument to the above we obtain that Q(√2) = a + √2b : a,b Q is a subfield of R. { ∈ } The following gives an example of a division ring, i.e. a unital ring in which every non-zero element has an inverse, but where multiplication is not necessarily commutative. Example 4.4. Let Q be the subset of all 2 2 complex matrices defined by × z w Q = : w,z C . w¯ z¯ ∈  −  

We claim that Q is a subring of M2(C). Let

z w x y p = and q = , w¯ z¯ y¯ x¯  −   −  for some complex numbers z,w,x,y. Then

z + x w + y z + x w + y p + q = = Q, w¯ y¯ z¯ +¯x w + y z + x ∈  − −   −  and zx wy¯ zy + wx¯ zx wy¯ zy + wx¯ pq = − = − Q, wx¯ z¯y¯ wy¯ +¯zx¯ zx wy¯ zy + wx¯ ∈  − − −   − −  using Lemma 3.1 (the additive and multiplicative properties of complex conjugation), so matrix addition and multiplication are operations on Q. It is also easy to see that

z w p = − − Q, − w¯ z¯ ∈  − 

13 so Q is indeed a subring of M2(C). An interesting property of Q is the following:

z w z¯ w zz¯ + ww¯ 0 1 0 = = (zz¯ + ww¯) . w¯ z¯ w¯ − z 0 zz¯ + ww¯ 0 1  −       Note that zz¯ + ww¯ = z 2 + w 2 is a positive real number as long as p is not the all-zero matrix. We get the same result if we perform| | the| product| in the reverse direction, which means that p has an inverse in Q, and this is given by − 1 z¯ w p 1 = − . ( z 2 + w 2) w¯ z | | | |   At this stage it is tempting to think that Q is a field, until we remember that matrix multiplication is not commutative, for example, the matrices

i 0 1 i and 0 i i 1  −    do not commute. Q is close to satisfying the axioms of a field, the only property it lacks is commutativity of multiplication. Rings of this type are called skew fields or division rings.

The matrix ring Q described above is actually the set of quaternions. This division ring was discovered by Hamilton in 1843, and will be discussed in the next section.

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