No. For example, z = -1 -mi is in C, but Re(z) < 0. Note that for a given value of m, C is a straight line through the origin. Which field axioms prevent this subspace from being a field?lolittaFarhat said:Homework Statement
Show that the only m for which the subspace of C given by {z ∈ C: Im(z) = m Re(z)} is a field is m=0.
Homework Equations
Field axioms
The Attempt at a Solution
I tried to prove one direction :
- If z is in the subspace, Re z>0
lolittaFarhat said:and m≠0 then Arg z<Arg z^2, so z^2 is not in the subspace.
I need a hint to prove the other direction.
A subspace of C is a subset of the complex numbers that satisfies the properties of a vector space. This means that it is closed under addition and scalar multiplication, contains the zero vector, and has closure under vector addition and scalar multiplication.
In order to prove that a subspace of C is a field, you must demonstrate that it satisfies the properties of a field. This includes commutativity, associativity, existence of additive and multiplicative inverses, and distributivity. You must also prove that it is closed under multiplication and that the multiplicative identity is contained within the subspace.
No, a subspace of C cannot be a field if it does not contain the multiplicative identity. The multiplicative identity is a necessary property of a field and without it, the subspace cannot be considered a field.
Proving that a subspace of C is a field is important because it shows that the subset of complex numbers satisfies the properties of a field, which is a fundamental concept in mathematics. This proof can also be used to show that other subsets of C are fields, and it can be applied to other vector spaces as well.
There are several methods that can be used to prove that a subspace of C is a field. These include proving that it satisfies the properties of a field, showing that it is isomorphic to a known field, or using the fact that every subspace of a field is also a field. Other methods may also be used depending on the specific properties of the subspace in question.