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### Linear Independence in Quotients of Polynomial Rings

Consider the following:

K = GF(2)    # can be an arbitrary field
R.<a,b> = PolynomialRing(K)
I = R.ideal(a^2-1,a*b)
Q = R.quo(I)


Now $1,a,b$ are linearly dependent in Q because $b=0$ (in Q).

Is there an elegant way to check this with Sage, especially when the ideals get more complicated? That is: Is there a method to check whether $(1,a_1,\dots,a_n)$ are linearly (in)dependent in a given quotient $K[a_1,\dots,a_n]/I$?

### Linear Independence in Quotients of Polynomial Rings

Consider the following:

K = GF(2)    # can be an arbitrary field
R.<a,b> = PolynomialRing(K)
I = R.ideal(a^2-1,a*b)
Q = R.quo(I)


Now $1,a,b$ are linearly dependent in Q because $b=0$ (in Q).

Is there an elegant way to check this with Sage, especially when the ideals get more complicated? That is: Is there a method to check whether $(1,a_1,\dots,a_n)$ $1,a_1,\dots,a_n$ are linearly (in)dependent in a given quotient $K[a_1,\dots,a_n]/I$?

 3 retagged FrédéricC 5141 ●3 ●43 ●112

### Linear Independence in Quotients of Polynomial Rings

Consider the following:

K = GF(2)    # can be an arbitrary field
R.<a,b> = PolynomialRing(K)
I = R.ideal(a^2-1,a*b)
Q = R.quo(I)


Now $1,a,b$ are linearly dependent in Q because $b=0$ (in Q).

Is there an elegant way to check this with Sage, especially when the ideals get more complicated? That is: Is there a method to check whether $1,a_1,\dots,a_n$ are linearly (in)dependent in a given quotient $K[a_1,\dots,a_n]/I$?