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1) Z12/1 is not a Field Always because if we take the ideal I = Z₁2/1 is a Field. 2) The map y: Z₁- ---Z₂ such that y(x) = 0 if x is even 1 if x is odd is not a ring homomorphism because 3) Eisenstin Criteria for irreducibility Test Fails for f(x)=x+ 5x³ -15x¹+ 15x³+25x² +5x+25 because but for p= f(x) is irreducible using mod p-test. 4) In a ring R; The sum of two non-trivial idempotent elements is not always an idempotent because in the ring if we take + is not 3 idempotent 5) There are more than two idempotent elements in the ring Z6OZ6; here are some of them (, ) (, ), (, ), (, ) 2 6) There is a multiplicative inverse for (2x+3) in Z₁[x] because (ax+3b) (2x+3)=1 where A = and b = 7) There is no proper non-trivial maximal ideals in (Z2₁, , ) is a False statement because < > is a maximal ideal in Z2₁, , ) 8) If (1+x) is an idempotent in Zn then x is Always an idempotent is a False statement because if x= 1+x is an idempotent element but x is not. " then

1) Z12/1 is not a Field Always because if we take the ideal I = Z₁2/1 is a Field. 2) The map y: Z₁- ---Z₂ such that y(x) = 0 if x is even 1 if x is odd is not a ring homomorphism because 3) Eisenstin Criteria for irreducibility Test Fails for f(x)=x+ 5x³ -15x¹+ 15x³+25x² +5x+25 because but for p= f(x) is irreducible using mod p-test. 4) In a ring R; The sum of two non-trivial idempotent elements is not always an idempotent because in the ring if we take + is not 3 idempotent 5) There are more than two idempotent elements in the ring Z6OZ6; here are some of them (, ) (, ), (, ), (, ) 2 6) There is a multiplicative inverse for (2x+3) in Z₁[x] because (ax+3b) (2x+3)=1 where A = and b = 7) There is no proper non-trivial maximal ideals in (Z2₁, , ) is a False statement because < > is a maximal ideal in Z2₁, , ) 8) If (1+x) is an idempotent in Zn then x is Always an idempotent is a False statement because if x= 1+x is an idempotent element but x is not. " then

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.4: Zeros Of A Polynomial
Problem 21E: Use Theorem to show that each of the following polynomials is irreducible over the field of...
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pleaseeeeeee solve question 3

1) Z₁2/1 Z₁2/1 is a Field. So if x is even 2) The map y: Z, -----→ Z₂ such that y(x) = { 1 if x is odd is not a ring homomorphism because 3) Eisenstin Criteria for irreducibility Test Fails for f(x)=x+ 5x³-15x¹+ 15x³+25x² +5x+25 because but for p= ,f(x) is irreducible using mod p-test. 4) In a ring R; The sum of two non-trivial idempotent elements is not always an idempotent because in the ring. idempotent if we take is not " 5) There are more than two idempotent elements in the ring Z6Z6; here are some of them (, ) , (, ) , ( , ) , (, ) 6) There is a multiplicative inverse for (2x+3) in Z₁[x] because (ax+3b) (2x+3)=1_where A = and b = 7) There is no proper non-trivial maximal ideals in (Z2₁, , ) is a False statement because < > is a maximal ideal in Z2₁, , ) 8) If (1+x) is an idempotent in Z, then x is Always an idempotent is a False statement because if x= 1+x is an idempotent element but x is not. 3 is not a Field Always because if we take the ideal I = then
Transcribed Image Text:1) Z₁2/1 Z₁2/1 is a Field. So if x is even 2) The map y: Z, -----→ Z₂ such that y(x) = { 1 if x is odd is not a ring homomorphism because 3) Eisenstin Criteria for irreducibility Test Fails for f(x)=x+ 5x³-15x¹+ 15x³+25x² +5x+25 because but for p= ,f(x) is irreducible using mod p-test. 4) In a ring R; The sum of two non-trivial idempotent elements is not always an idempotent because in the ring. idempotent if we take is not " 5) There are more than two idempotent elements in the ring Z6Z6; here are some of them (, ) , (, ) , ( , ) , (, ) 6) There is a multiplicative inverse for (2x+3) in Z₁[x] because (ax+3b) (2x+3)=1_where A = and b = 7) There is no proper non-trivial maximal ideals in (Z2₁, , ) is a False statement because < > is a maximal ideal in Z2₁, , ) 8) If (1+x) is an idempotent in Z, then x is Always an idempotent is a False statement because if x= 1+x is an idempotent element but x is not. 3 is not a Field Always because if we take the ideal I = then
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