Understand the problem

Find the number of ring homomorphism from $\Bbb Z[x,y]$ to $\Bbb F_2[x]/(x^3+x^2+x+1)$

Source of the problem
TIFR 2019 GS Part A, Problem 16
Algebra
Moderate
Suggested Book
Abstract Algebra, Dummit and Foote

Do you really need a hint? Try it first!

Now observe that $(x^3+x^2+x+1)=(x+1)^3$ in $\Bbb F_2[x]$. Then can you think about the ring $\Bbb F_2[x]/(x^3+x^2+x+1)=\Bbb F_2[x]/(x+1)^3$?

Then we have to find the number of ring homomorphism from $\Bbb Z[x,y]$ to $\Bbb F_2[x]/(x^3+x^2+x+1)=\Bbb F_2[x]/(x+1)^3$. How many elements does the ring have? $\Bbb F_2[x]/(x^3+x^2+x+1)=\Bbb F_2[x]/(x+1)^3\neq\Bbb F_2$. It is not a field but it has $8$ elements as $\Bbb F_2[x]/(x^3+x^2+x+1)=\Bbb F_2[x]/(x+1)^3=\{a+bx+cx^2: a,b,c \in \Bbb F_2\}$.

Now what is the next move?

For any commutative ring $R$ there is a bijection between the homomorphisms $\phi:\mathbb{Z}[x,y]\to R$ and the elements in $R^2$. To any $(a,b)\in R^2$, $\phi(x)=a$ and $\phi(y)=b$ determines the homomorphism.

Hence, if one counts the number of elements in $\mathbb{F}_2[x]/(x^3+x^2+x+1)$, the number of homomorphisms you want is just the square.

Now, $Bbb F_2[x]/(x+1)^3=\{a+bx+cx^2: a,b,c \in \Bbb F_2\}$ has $2^3$ elements.

Connected Program at Cheenta

College Mathematics Program

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

Similar Problems

Rational maps to irrational and vice versa?: TIFR 2019 GS Part B, Problem 1

It is an analysis question on functions. It was asked in TIFR 2019 GS admission paper.

Expected expectation:TIFR 2019 GS Part A, Problem 20

It is an statistics question on the probability and expectation. It was asked in TIFR 2019 GS admission paper.

Functions on differential equation:TIFR 2019 GS Part A, Problem 19

It is an analysis question on the differential equation. It was asked in TIFR 2019 GS admission paper.

Permutation or groups!: TIFR 2019 GS Part A, Problem 18

It is an combinatorics question on the permutation or application of counting principle. It was asked in TIFR 2019 GS admission paper.

Compact or connected:TIFR 2019 GS Part A, Problem 17

It is a topology question on the compactness and connectedness. It was asked in TIFR 2019 GS admission paper.

Finding the number of ring homomorphisms:TIFR 2019 GS Part A, Problem 16

It is an algebra question on the ring homomorphism. It was asked in TIFR 2019 GS admission paper.

Uniform Covergence:TIFR 2019 GS Part A, Problem 15

It is an analysis question on the uniform convergence of sequence of functions. It was asked in TIFR 2019 GS admission paper.

Problems on continuity and differentiability:TIFR 2019 GS Part A, Problem 14

It is an analysis question on the continuity and differentiability of functions. It was asked in TIFR 2019 GS admission paper.

Iterative sequences:TIFR 2019 GS Part A, Problem 13

It is an analysis question on the sequence of iterations. It was asked in TIFR 2019 GS admission paper.

Sequence of functions:TIFR 2019 GS Part A, Problem 12

It is an analysis question on the sequence of functions. It was asked in TIFR 2019 GS admission paper.