# Ernst Kummer

> German mathematician (1810–1893)

**Wikidata**: [Q57245](https://www.wikidata.org/wiki/Q57245)  
**Wikipedia**: [English](https://en.wikipedia.org/wiki/Ernst_Kummer)  
**Source**: https://4ort.xyz/entity/ernst-kummer

## Summary
Ernst Kummer was a German mathematician (1810–1893) known for his foundational work in number theory, particularly his development of ideal numbers and his proof of Fermat's Last Theorem for regular primes. He was a professor at the University of Breslau (now Wrocław) and made significant contributions to algebraic geometry and the theory of quadratic forms.

## Biography
- Born: January 29, 1810, in Sorau, Kingdom of Prussia (now Żary, Poland)
- Nationality: German
- Education: Studied at the University of Breslau (now Wrocław) and the University of Berlin
- Known for: Pioneering work in number theory, including the concept of ideal numbers and proofs related to Fermat's Last Theorem
- Employer(s): University of Breslau (1833–1893)
- Field(s): Number theory, algebraic geometry

## Contributions
- **Ideal Numbers**: Introduced the concept of ideal numbers in algebraic number theory, which became a cornerstone of modern algebraic geometry.
- **Fermat's Last Theorem**: Proved the theorem for regular primes, contributing to the broader understanding of the problem.
- **Quadratic Forms**: Made significant advances in the theory of quadratic forms, particularly in the context of algebraic number fields.
- **Kummer Theory**: Developed a mathematical theory describing field extensions involving the adjunction of nth roots, named after him.
- **Kummer's Test**: Formulated a theorem used in analysis to determine the convergence of series.

## FAQs
### What was Ernst Kummer's most significant contribution to mathematics?
Ernst Kummer's most significant contribution was the introduction of ideal numbers in algebraic number theory, which provided a framework for understanding algebraic structures that are not immediately apparent in classical number theory.

### Where did Ernst Kummer conduct his academic work?
Ernst Kummer spent his entire academic career at the University of Breslau (now Wrocław), where he held a professorship from 1833 until his death in 1893.

### What was Kummer's approach to Fermat's Last Theorem?
Kummer proved Fermat's Last Theorem for regular primes, which involved developing a theory of ideal numbers to handle cases where the theorem did not hold in classical terms.

### How did Kummer's work influence algebraic geometry?
Kummer's work on ideal numbers and field extensions laid the groundwork for modern algebraic geometry, influencing later developments in the field.

### What was Kummer's educational background?
Kummer studied at the University of Breslau and later at the University of Berlin, where he was influenced by prominent mathematicians of his time.

## Why They Matter
Ernst Kummer's work in number theory revolutionized algebraic geometry and provided a deeper understanding of algebraic structures. His concept of ideal numbers resolved inconsistencies in classical number theory and paved the way for modern algebraic geometry. Kummer's proofs related to Fermat's Last Theorem, though not a complete solution, demonstrated the power of algebraic methods in solving fundamental problems. His contributions to the theory of quadratic forms and his development of Kummer theory have had lasting impacts on mathematical research. Kummer's legacy continues to influence number theory and algebraic geometry, making him a foundational figure in these fields.

## Notable For
- **Pioneer in Ideal Numbers**: Introduced the concept of ideal numbers, which became essential in algebraic number theory.
- **Proof of Fermat's Last Theorem for Regular Primes**: Provided a significant partial solution to Fermat's Last Theorem.
- **Advancements in Quadratic Forms**: Made key contributions to the theory of quadratic forms in algebraic number fields.
- **Development of Kummer Theory**: Created a mathematical theory describing field extensions involving the adjunction of nth roots.
- **Kummer's Test**: Formulated a theorem used in analysis to determine the convergence of series.
- **Longtime Professor at University of Breslau**: Spent his entire academic career at the University of Breslau, shaping mathematical education in Prussia.

## Body

### Early Life and Education
Ernst Kummer was born on January 29, 1810, in Sorau, Kingdom of Prussia (now Żary, Poland). He studied at the University of Breslau, where he was influenced by the mathematical environment, and later at the University of Berlin. His early education laid the foundation for his later work in number theory and algebraic geometry.

### Academic Career
Kummer began his academic career at the University of Breslau in 1833, where he remained for the rest of his life. He held the position of professor and made significant contributions to mathematics during his tenure. His work at the university was marked by his pioneering research in number theory and algebraic geometry.

### Contributions to Number Theory
Kummer's most notable contribution to number theory was the introduction of ideal numbers. This concept resolved inconsistencies in classical number theory and provided a framework for understanding algebraic structures that were not immediately apparent. His work on ideal numbers became a cornerstone of modern algebraic geometry.

### Proof of Fermat's Last Theorem for Regular Primes
Kummer proved Fermat's Last Theorem for regular primes, which involved developing a theory of ideal numbers to handle cases where the theorem did not hold in classical terms. This partial solution demonstrated the power of algebraic methods in solving fundamental problems in number theory.

### Advancements in Quadratic Forms
Kummer made significant advances in the theory of quadratic forms, particularly in the context of algebraic number fields. His work in this area provided a deeper understanding of the properties and behaviors of quadratic forms, influencing later developments in the field.

### Development of Kummer Theory
Kummer developed a mathematical theory describing field extensions involving the adjunction of nth roots. This theory, named after him, has had a lasting impact on algebraic number theory and has been used to solve various problems in the field.

### Kummer's Test
Kummer formulated a theorem used in analysis to determine the convergence of series. This theorem, known as Kummer's Test, is a fundamental tool in mathematical analysis and has been widely applied in various areas of mathematics.

### Legacy and Influence
Ernst Kummer's legacy continues to influence number theory and algebraic geometry. His work on ideal numbers and field extensions has been foundational in these fields, and his proofs related to Fermat's Last Theorem have demonstrated the power of algebraic methods. Kummer's contributions to the theory of quadratic forms and his development of Kummer theory have had lasting impacts on mathematical research. His work at the University of Breslau shaped mathematical education in Prussia and influenced generations of mathematicians.

## References

1. Czech National Authority Database
2. Great Soviet Encyclopedia (1969–1978)
3. BnF authorities
4. Integrated Authority File
5. MacTutor History of Mathematics archive
6. [Source](https://www.math.berlin/mathematiker/ernst-eduard-kummer.html)
7. [MacTutor History of Mathematics archive](http://www-history.mcs.st-andrews.ac.uk/Biographies/Kummer.html)
8. Mathematics Genealogy Project
9. Complete List of Royal Society Fellows 1660-2007
10. [Mathematics Genealogy Project](http://www.genealogy.ams.org/id.php?id=18336)
11. International Standard Name Identifier
12. Virtual International Authority File
13. CiNii Research
14. [Source](https://kalliope-verbund.info/DE-611-BF-116360)
15. Ernst Kummer. SNAC
16. Geneological database Merkel-Zeller
17. Brockhaus Enzyklopädie
18. Croatian Encyclopedia
19. Freebase Data Dumps. 2013
20. [Source](https://books.google.cat/books?id=tqaWlHIsZXAC&pg=PA14)
21. [Source](http://digitale.beic.it/primo_library/libweb/action/search.do?fn=search&vid=BEIC&vl%283134987UI0%29=creator&vl%28freeText0%29=Kummer%20Ernst%20Eduard)
22. Treccani's Enciclopedia on line
23. Enciclopedia Treccani