TIFR 2014 Problem 26 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Linear Algebra Done Right by Sheldon Axler. This book is very useful for the preparation of TIFR Entrance.
Also Visit: College Mathematics Program of Cheenta
The number of irreducible polynomials of the form (x^2+ax+b) , with (a,b) in the field (\mathbb{F}_7) of 7 elements is:
A. 7
B. 21
C. 35
D. 49
First, what is the number of polynomials of the form (x^2+ax+b) in (\mathbb{F}_7) ? (a) has 7 choices and (b) has 7 choices. So we have a total of (7 \times 7 =49) monic degree 2 polynomials in (\mathbb{F}_7).
How many of these are reducible? If a monic polynomial of degree 2 is reducible then it must break into two factors in the form (p(x)=(x-\alpha)(x-\beta) ).
We want to count the number of polynomials of the form (p(x)=(x-\alpha)(x-\beta)). This will give us the number of reducible polynomials and we will subtract it from the total number to get the number of irreducible.
For the counting purposes, we must be careful not to have repetitions. For (\alpha=0) we can allow (\beta=0,1,..,6). For (\alpha=1) we can allow (\beta=1,...,6) (NOT 0 because that will again give ((x-0)(x-0)) which we have already counted in the (\alpha=0) situation ).
For (\alpha=2) we have (\beta=2,...,6) and so on... for (\alpha=6) we have (\beta=6) only.
So in total, we have (7+6+...+1=\frac{7\times 8}{2} = 28 ) polynomials which are reducible, has degree 2 and is monic.
Therefore, the number of irreducible polynomials are (49-28=21).
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.
