0% Complete
0/59 Steps

# 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

# Start with hints

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

## Matrix additive-multiplicative :TIFR 2018 Part A Problem 19

This problem is a cute and simple application of additive and multiplicative properties of matrices in the linear algebra section. It appeared in TIFR GS 2018.

## Commutative does not commute in matrices: TIFR 2018 Part A, Problem 11

This problem is a cute and simple approach using beautiful fact of matrices transformation in the linear algebra section. a, b and c can be taken from any commutative ring with identity, often taken to be the ring of real numbers or the ring of integers .It appeared in TIFR GS 2018.

## Spanning matrix space by niltopent matrices: TIFR 2018 Part A, Problem 15

This problem is a cute and simple application of spanning matrix space by niltopent matrices in the linear algebra section. It appeared in TIFR GS 2018.

## Calculating eigenvalues through geometry:TIFR 2018 Part A, Problem 14

This problem is a cute and simple application of calculating eigenvalues through geometry in the linear algebra section. It appeared in TIFR GS 2018.

## Direct product of groups: TIFR 2018 Part A, Problem 9

This problem is a cute and simple application of the direct product of groups in the abstract algebra section. It appeared in TIFR GS 2018.

## Real Symmetric matrix: TIFR 2018 Part A, Problem 6

This problem is a cute and simple application of the real symmeric matrices in the linear algebra section. It appeared in TIFR GS 2018.

## Integration of nonnegative function: TIFR 2018 Part A, Problem 4

This problem is a cute and simple application of the integration of nonnegative function in the analysis section. It appeared in TIFR GS 2018.

## Application of L’Hopital: TIFR GS 2018 Part A, Problem 2

This problem is a cute and simple application of the L’Hopital rule in the analysis section. It appeared in TIFR GS 2018.

## I.S.I. M.Math Entrance 2019 (PG) Solutions, Hints, and Answer Key

This is a work in progress. Please suggest improvements in the comment section.Problems Objective I.S.I. M.Math Subjective 2019 Objective Section (Answer Key)  1. C 2. D 3. B 4. C 5. A 6. D 7. B 8. D 9. D 10. D 11. A 12. B 13. B 14. A 15. C 16. A 17. C 18. A 19....

## Clocky Rotato Arithmetic

This problem is a cute and simple application of the real symmeric matrices in the linear algebra section. It appeared in TIFR GS 2018.