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Thus deg(f⋅g) = 0 which is not greater than the degrees of f and g (which each had degree 1). Since the norm function is not defined for the zero element of the ring, we consider the degree of the polynomial f(x) = 0 to also be undefined so that it follows the rules of a norm in a Euclidean domain.
Unlike other constant polynomials, its degree is not zero. Rather, the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or −∞). [10] The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of roots. The graph of the zero ...
Since it is not a proper extension, its degree is 1 and therefore the degree of p(x) is 1. On the other hand, if F has some proper algebraic extension K, then the minimal polynomial of an element in K \ F is irreducible and its degree is greater than 1.
This makes the zero polynomial useless for classifying different values of α into types, so it is excepted. If there are any non-zero polynomials in J α, i.e. if the latter is not the zero ideal, then α is called an algebraic element over F, and there exists a monic polynomial of least degree in J α.
In linear algebra, the minimal polynomial μ A of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P(A) = 0. Any other polynomial Q with Q(A) = 0 is a (polynomial) multiple of μ A. The following three statements are equivalent: λ is a root of μ A, λ is a root of the characteristic polynomial χ A ...
If the degree of the GCD is greater than i, then Bézout's identity shows that every non zero polynomial in the image of has a degree larger than i. This implies that S i = 0 . If, on the other hand, the degree of the GCD is i , then Bézout's identity again allows proving that the multiples of the GCD that have a degree lower than m + n − i ...
In the special case of the zero polynomial, all of whose coefficients are zero, the leading coefficient is undefined, and the degree has been variously left undefined, [9] defined to be −1, [10] or defined to be a −∞. [11] A constant polynomial is either the zero polynomial, or a polynomial of degree zero.
The equation of a line in a Euclidean plane is linear, that is, it equates a polynomial of degree one to zero. So, the Bézout bound for two lines is 1, meaning that two lines either intersect at a single point, or do not intersect. In the latter case, the lines are parallel and meet at a point at infinity.