Poor, Smart, and Handsome, the Tale of a Mathematician...

Niels Henrik Abel

 

     

In Abstract Algebra, when we have a group where $ab = ba$ for all choices of group elements $a$ and $b$, we say the group is commutative, or Abelian, named in honor of Niels Abel.

Niels Abel was a young, Norwegian mathematician. He was born August 5, 1802 in Finnoy, Norway and passed away on April 6, 1829 in Froland, at the young age of 26. He died from tuberculosis. It has been said though that, "He [Abel] has left mathematicians something to keep them busy for five hundred years." - Charles Hermite

By the age of 16 Abel began reading mathematical works by Newton, Gauss, Lagrange, and Euler-introduced to him by his teacher Bernt Holmboe, who saw great potential in the young man.

Despite his father passing away in 1820, leaving his family in a worse financial state than they were before, Abel attended the University of Christiania in 1821.  This was thanks to Holmboe who was able to get Abel a scholarship. A year later Abel graduated. He continued to study mathematics on his own and through traveling to collaborate with others. 

Though still very poor and ill, often only able to afford one meal a day, Abel worked hard to print his work and send it to universities and other mathematicians.

Through his adventures, Abel was able to pick up friends who encouraged him in his work and helped out a bit financially. Among those was Bernt Holmboe who got him started in mathematics, a proffessor from college- Christopher Hansteen and his wife who cared for Abel like he was family, Christine Kemp who was Abel's fiancé, and August Leopold Crelle the creator of Crelle's Journal, a journal for mathematical papers. He was Abel's friend, helped print his work, and helped to get him an academic position working in Berlin, but Abel had passed away before the good news could reach him. 

At the time, few people recognized the influence and importance of Abel's work. However, today is a different story. In fact, in 2002 Norway created a prize called the Abel prize for mathematics in his honor. It is considered the equivalent to to a Nobel Prize. 

Among the many mathematical topics Abel studied and worked on, Abel's Impossibility Theorem stuck out to me the most. It states that,

    "The general algebraic equation with one unknown of degree greater than $4$ is insoluble in radicals, i.e. there does not exist a formula, which expresses the roots of a general equation of degree greater than four in terms of the coefficients involving the operations of addition, subtraction, multiplication, division, raising to a natural degree, and extraction of roots of natural degree."

To see a proof with some author notes to go along with the theorem, here is a link that walks you through the process http://www.math.caltech.edu/~jimlb/abel.pdf . Look at pg. 7-13.

So now we know, if anyone tries to ask us to solve an equation with a polynomial of degree five or higher, it may be a bit trickier than pulling out our quadratic formula. Thank you Abel! 

*Favorite Quote From Niels Abel, "Everyone wants to teach and no one wants to learn."

Thanks Bunches!

    -Haylee Jo Lau

Works Cited:

https://en.wikipedia.org/wiki/Niels_Henrik_Abel

https://www.britannica.com/biography/Niels-Henrik-Abel

http://www.math.wichita.edu/history/Men/abel.html

http://kobotis.net/math/MathematicalWorlds/Fall2016/131/Biography/Abel.pdf

https://mathshistory.st-andrews.ac.uk/Biographies/Abel/

https://mathigon.org/timeline/abel

http://www.math.caltech.edu/~jimlb/abel.pdf

https://arxiv.org/pdf/1908.00972.pdf

Contemporary Abstract Algebra, Joseph A. Gallian, Ninth Edition

    Here is my second post to test how well I can use Latex in my blog. Fingers crossed!!! 

(P.S. This is the first write up I turned in for my Abstract Algebra class, now there is something "to see" as my Padre would say.)

    Suppose you encounter some ciphertext $(i,q,x,x,p,a,z,q)$ that you know has been encrypted using a simple shift permutation $\phi_n$ for some $n$. 

$1$.  Decode the ciphertext and state the original message. State the value of $n$ that was used to encode the original message.

The message found after decoding the ciphertext said, $(w,e,l,l,d,o,n,e)$. The original message was encoded using $\phi_{12}$. We know this because $\phi_{12}(w)=i$ and so forth.

$2$.  Encode the plain text message "attack at dawn" into cipertext using the encryption key you discovered in the first part.

Using the encryption key we discovered in the first part, we would encode the message $(a,t,t,a,c,k,a,t,d,a,w,n)$ using $\phi_{12}$. In the end we get the new ciphertext $(m,f,f,m,o,w,m,f,p,m,i,z)$.

$3$.  Are there any other $n$ that would have produced the same results as above? Be ready to fully justify your answer.

There are other $n$ that would have produced the same results as above. Another shift permutation $\phi_n$ that would work is $\phi_{38}$. We know that this will work because $12 + 26 = 38$. We can see then, that $\phi_{38}$ works because we will shift right $26$ times and then an additional $12$ times landing at the same place we did when we just moved $\phi_{12}$.

 Well, this is the first post on my blog, the Faces Behind Great Mathematics- pretty exciting! This is mostly just a tester post for me to figure out what I am doing. :)

-Haylee Jo Lau