Understanding the Galois Groups of Real and Complex Numbers

When delving into the realm of abstract algebra, the study of fields and their automorphisms often leads us to the concept of Galois groups. These groups provide profound insights into the structure and relationships among different fields. In this article, we will explore the Galois groups for the reals over the rationals and for the complex numbers over the algebraic numbers. We will analyze their structure, size, and nature to understand the distinctions between these two groups.

The Galois Group of the Reals over the Rationals

The field extension R/Q involves the real numbers over the rational numbers. The Galois group Gal(R/Q) consists of field automorphisms of R that fix Q. While these automorphisms are numerous, only one non-trivial automorphism exists, which is the complex conjugation. This automorphism sends a complex number z to its conjugate zˉ.

Structure of the Galois Group

The structure of Gal(R/Q) can be described as a cyclic group of order 2, which we denote as Z/2Z. This means that the group is generated by the complex conjugation, which effectively swaps elements of the form a bi with a - bi, where a and b are rational numbers.

The Galois Group of the Complex Numbers over the Algebraic Numbers

The field extension C/Q? involves the complex numbers over the algebraic numbers, where Q? is the field generated by all algebraic numbers over Q. The Galois group Gal(C/Q?) consists of automorphisms of C that fix Q?. Given that C is algebraically closed, every polynomial with coefficients in Q? can be completely factored in C.

Structure of the Galois Group

The structure of Gal(C/Q?) is considerably more complex. It is an uncountable Galois group and can be described as a profinite group. The profinite structure reflects the intricate nature of the automorphisms, which permute the roots of polynomials with coefficients in Q?. This group encapsulates the rich structure of permutations over the algebraic roots, making it significantly more vast and complex than its counterpart for the reals.

Summary of Differences

Size

One of the most striking differences is the size of the Galois groups. The Galois group of the reals over the rationals, Gal(R/Q), is finite, specifically of order 2. In contrast, the Galois group of the complex numbers over the algebraic numbers, Gal(C/Q?), is uncountably infinite. This vast difference in size reflects the complexity of automorphisms in the two fields.

Nature of the Groups

The nature of the Galois groups also differs significantly. The Galois group of the reals over the rationals is relatively simple, consisting solely of the complex conjugation. On the other hand, the Galois group of the complex numbers over the algebraic numbers is complex and involves a rich structure of permutations. These permutations are determined by the roots of polynomials with coefficients in the algebraic field, reflecting its intricate nature.

Field Characteristics

Another key difference lies in the field characteristics. The field of reals, R, is not algebraically closed, which restricts the types of automorphisms that can be present. In contrast, the field of complex numbers, C, is algebraically closed, leading to a much broader set of automorphisms. This difference in algebraic closure is essential in understanding the structure of the Galois groups.

Understanding these groups provides deep insights into the relationships between different fields in algebra and number theory, making them fundamental concepts in these areas of study. By grasping the distinctions between these Galois groups, we can better comprehend the nature of mathematical structures and the relationships between them.