Aleksandr Gelfond

Definition

Aleksandr Gelfond was a prominent Soviet mathematician known for his groundbreaking work in transcendental number theory and for establishing the transcendence of certain numbers, notably $$e^a$$ where $$a$$ is algebraic and not zero. His contributions are fundamental in understanding the nature of transcendental numbers, particularly in relation to the famous constants $$ ext{π}$$ and $$e$$.

5 Must Know Facts For Your Next Test

1. Gelfond proved in 1934 that the number $$e^{ ext{π}}$$ is transcendental, which is a major result in transcendental number theory.
2. He was awarded the Stalin Prize in 1946 for his contributions to mathematics, highlighting the significance of his work during that era.
3. Gelfond's results on transcendence opened up new avenues for research in number theory and had lasting impacts on mathematics as a whole.
4. The Gelfond–Schneider theorem is one of the cornerstones in the field of transcendental numbers, demonstrating a specific case of how algebraic and transcendental numbers interact.
5. His work laid essential groundwork for later mathematicians exploring complex numbers and their properties.

Review Questions

• What impact did Gelfond's work have on our understanding of transcendental numbers?
  • Gelfond's work significantly advanced our understanding of transcendental numbers by proving the transcendence of numbers like $$e^{ ext{π}}$$. This revelation not only confirmed the existence of more complex relationships between algebraic and transcendental numbers but also provided a solid foundation for further exploration into number theory. His contributions underscored the complexity and richness of mathematical constants such as $$e$$ and $$ ext{π}$$.
• Discuss the implications of the Gelfond–Schneider theorem on the field of mathematics.
  • The Gelfond–Schneider theorem has profound implications for the field of mathematics as it establishes a clear connection between algebraic and transcendental numbers. By demonstrating that certain exponentials of algebraic numbers yield transcendental results, it deepened the understanding of how different types of numbers interact. This theorem paved the way for future investigations into number theory and enhanced our grasp of mathematical properties related to irrationality and transcendence.
• Evaluate how Gelfond's discoveries have influenced modern mathematical theories related to transcendental numbers.
  • Gelfond's discoveries have been pivotal in shaping modern mathematical theories surrounding transcendental numbers. His proof regarding the transcendence of $$e^{ ext{π}}$$ not only settled long-standing questions but also inspired subsequent research into other mathematical constants. As mathematicians continue to explore deeper connections between algebraic structures and transcendence, Gelfond's work remains foundational, influencing both theoretical perspectives and practical applications in fields such as cryptography and numerical analysis.

Related terms

Transcendental Numbers: Numbers that are not roots of any non-zero polynomial equation with rational coefficients, such as $$ ext{π}$$ and $$e$$.

Gelfond–Schneider Theorem: A theorem stating that if $$a$$ is algebraic and not equal to 0 or 1, and $$b$$ is an algebraic number that is not a rational number, then $$a^b$$ is transcendental.

Lindemann–Weierstrass Theorem: A theorem stating that if $$α_1, α_2, ext{...,} α_n$$ are distinct algebraic numbers, then $$e^{α_1}, e^{α_2}, ext{...,} e^{α_n}$$ are linearly independent over the algebraic numbers.