Freshman's dream#Prime characteristic

• $(1+4)^2\; =\; 5^2\; =\; 25$, but $1^2+4^2\; =\; 17$.
• $\backslash sqrt\{x^2+y^2\}$ does not equal $\backslash sqrt\{x^2\}+\backslash sqrt\{y^2\}=|x|+|y|$. For example, $\backslash sqrt\{9+16\}=\backslash sqrt\{25\}=5$, which does not equal {{nowrap|1=3 + 4 = 7}}. In this example, the error is being committed with the exponent {{nowrap|1=n = {{sfrac|1|2}}}}.

When $p$ is a prime number and $x$ and $y$ are members of a commutative ring of characteristic $p$, then $(x+y)^p=x^p+y^p$. This can be seen by examining the prime factors of the binomial coefficients: the nth binomial coefficient is

:$\backslash binom\{p\}\{n\}\; =\; \backslash frac\{p!\}\{n!(p-n)!\}.$

The numerator is p factorial(!), which is divisible by p. However, when {{nowrap|0 < n < p}}, both n! and {{nowrap|(p − n)!}} are coprime with p since all the factors are less than p and p is prime. Since a binomial coefficient is always an integer, the nth binomial coefficient is divisible by p and hence equal to 0 in the ring. We are left with the zeroth and pth coefficients, which both equal 1, yielding the desired equation.

Thus in characteristic p the freshman's dream is a valid identity. This result demonstrates that exponentiation by p produces an endomorphism, known as the Frobenius endomorphism of the ring.

The demand that the characteristic p be a prime number is central to the truth of the freshman's dream. A related theorem states that if p is prime then {{nowrap|(x + 1)^p ≡ x^p + 1}} in the polynomial ring $\backslash mathbb\{Z\}\_p[x]$. This theorem is a key fact in modern primality testing.A. Granville, [https://www.ams.org/bull/2005-42-01/S0273-0979-04-01037-7/S0273-0979-04-01037-7.pdf It Is Easy To Determine Whether A Given Integer Is Prime], Bull. of the AMS, Volume 42, Number 1 (Sep. 2004), Pages 3–38.

In 1938, Harold Willard Gleason published a poem titled «"Dark and Bloody Ground---" (The Freshman's Dream)» in The New York Sun on September 6, which was subsequently reprinted in various other newspapers and magazines. It consists of 2 stanzas, each containing 8 lines with alternating indentation; it has an ABCB rhyming scheme. Words and phrases that hint that it might be related to this concept include: "Algebra", "Wild corollaries twine", "surds", "of plus and minus sign", "binomial", "quadratic", "parenthesis", "exponents", "in terms of x and y", "remove the brackets, radicals, and do so with discretion", and "factor cubes".Original source: {{Citation |first=Harold Willard |last=Gleason |title="Dark and Bloody Ground---" (The Freshman's Dream) |date=September 6, 1938 |newspaper=The New York Sun }}. Reproduced in: {{#invoke:String|replace|pattern=|replace=|source=

• {{Citation |no-tracking=y|date=September 9, 1938 |newspaper=The Ottawa Journal |publication-place=Ottawa, Ontario, Canada |volume=LIII |issue=229 |page=6 |id=Newspapers.com [https://www.newspapers.com/image/45993022/?match=1&terms=%22freshman%27s%20dream%22%20algebra 45993022]}}
• {{Citation |no-tracking=y|date=September 15, 1938 |newspaper=The Moncton Transcript |publication-place=Moncton, Moncton Parish, New Brunswick, Canada |volume=LVII |issue=92 |page=[4] |id=Newspapers.com [https://www.newspapers.com/image/1104615164/?terms=%22freshman%27s%20dream%22%20algebra&match=1 1104615164] }}
• {{Citation |no-tracking=y|date=September 29, 1938 |newspaper=The Boston (Daily) Globe |publication-place=Boston, Massachusetts, United States |volume=CXXXIV |issue=91 |page=28 |id=Newspapers.com [https://www.newspapers.com/image/431880838/?match=1&terms=%22freshman%27s%20dream%22%20algebra 431880838], [https://www.newspapers.com/image/431883159/?match=1&terms=%22freshman%27s%20dream%22%20algebra 431883159], {{ProQuest|847896830}}}}
• {{Citation|no-tracking =y|date=October 1938 |journal=National Mathematics Magazine |volume=13 |issue=1 |page=50 |publisher=Mathematical Association of America |issn=1539-5588 |jstor=3028376 |title="Dark and Bloody Ground---" |last1=Gleason |first1=Harold Willard | author-mask=0}}.}}

The history of the term "freshman's dream" is somewhat unclear. In a 1940 article on modular fields, Saunders Mac Lane quotes Stephen Kleene's remark that a knowledge of {{nowrap|1=(a + b)² = a² + b²}} in a field of characteristic 2 would corrupt freshman students of algebra. This may be the first connection between "freshman" and binomial expansion in fields of positive characteristic.Colin R. Fletcher, Review of Selected papers on algebra, edited by Susan Montgomery, Elizabeth W. Ralston and others. Pp xv, 537. 1977. {{ISBN|0-88385-203-9}} (Mathematical Association of America), The Mathematical Gazette, Vol. 62, No. 421 (Oct., 1978), The Mathematical Association. p. 221. Since then, authors of undergraduate algebra texts took note of the common error. The first actual attestation of the phrase "freshman's dream" seems to be in Hungerford's graduate algebra textbook (1974), where he states that the name is "due to" Vincent O. McBrien.Thomas W. Hungerford, Algebra, Springer, 1974, [https://www.google.com/books/edition/Algebra/KvruAAAAMAAJ?hl=en&gbpv=1&bsq=McBrien p. 121 (with McBrien's name also stated on pp. ix and 498)]; also in Abstract Algebra: An Introduction, 2nd edition. Brooks Cole, July 12, 1996, p. 366. Alternative terms include "freshman exponentiation", used in Fraleigh (1998).John B. Fraleigh, A First Course In Abstract Algebra, 6th edition, Addison-Wesley, 1998. pp. 262 and 438. The term "freshman's dream" itself, in non-mathematical contexts, is recorded since the 19th century.[http://www.google.com/search?tbo=p&tbm=bks&q=%22freshman%27s+dream%22&tbs=,cdr:1,cd_min:Jan%201_2%201800,cd_max:Dec%2031_2%201900&num=10 Google books 1800–1900 search for "freshman's dream"]: [https://books.google.com/books?id=3XNHAAAAYAAJ&pg=PA176&dq=%22freshman%27s+dream%22 Bentley's miscellany, Volume 26, p. 176], 1849

Since the expansion of {{nowrap|1=(x + y)ⁿ}} is correctly given by the binomial theorem, the freshman's dream is also known as the "child's binomial theorem" or "schoolboy binomial theorem".

• Pons asinorum
• Primality test
• Sophomore's dream
• Frobenius endomorphism

{{reflist|2}}

Category:Algebra education

Category:Mathematical fallacies

Category:Theorems in ring theory

Category:Prime numbers